Sessions 1&2: Course Overview and Introduction to Derivatives
Futures and Options
Course number:
FINC-UB.0043 Futures and Options
Course description: This course is designed to introduce Finance students to the theoretical and real
world aspects of financial futures, options, and other derivatives. Over the last 40 years, the markets for
these versatile instruments have grown enormously and have generated a profusion of innovative products
and ideas, not to mention periodic crises. Derivatives have become one of the most important tools of
modern finance, from both the academic and the practical standpoint. The subject is inherently more
quantitative than other business courses, but the emphasis in this course is not on the math and theory, but
always on developing your intuition. The goal is for you to understand the principles of how these
instruments and markets work, not to derive models and prove theorems.
Professor:
Email:
Telephone:
Stephen Figlewski
sfiglews@stern.nyu.edu
212-998-0712
Office:
Office hours:
KMEC 9-64
Tuesday/Thursday after class (5:00-6:30); and by appointment
Course website:
Course materials and announcements will be posted on the course
website.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
1
Part 2, Session 1: Ramping up to Black-Scholes
This session begins the second portion of the course, which focuses on option
valuation theory and practice and more specialized "advanced topics."
The development of option theory is one of the major triumphs of modern finance.
It has contributed to enormous growth and proliferation of trading in derivative
securities, as well as the use of new theoretical valuation tools for derivatives
valuation and risk management
In 1997 the Nobel Prize was awarded to Myron Scholes and Robert Merton in
recognition of the importance of the option pricing model. Fischer Black would
surely have shared the award, but he unfortunately died in 1995, before it was
given.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
2
Put-Call Parity and Option Properties from Portfolio Dominance
A Very Important Arbitrage Trade
Consider a call and a put on XYZ stock, with the same maturity, 1 month, and the same
strike price, 100. (Both options are European.)
If XYZ is above 100 in a month, the 100 strike call will be exercised.
If XYZ is below 100 in a month, the 100 strike put will be exercised .
What if you buy the call and also write the put?
• Suppose XYZ ends up at 105. You choose to exercise the call: you pay 100 and
you get the stock. (The put is out of the money and expires without being
exercised.)
• What if XYZ is at 95? The put will be exercised by your counterparty and you
will have to buy the stock for 100. (The call is worthless.)
In other words, no matter what happens with the stock price, at maturity you will pay 100
and you will own the stock.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
3
Put-Call Parity and Option Properties from Portfolio Dominance
Put-Call Parity
This position returns exactly what you would have if you just bought the stock today for S =
100, and borrowed the present value of 100 to finance a large part of the position. (On option
maturity day, you would own the stock and pay out 100.)
With two different ways to achieve exactly the same payoff, there is a possible arbitrage
trade. If these two positions do not cost exactly the same to set up, arbitrageurs will buy the
position in the way that is cheaper and sell it (i.e., take the opposite position) in the way that
is more expensive. The difference in cost between them is an arbitrage profit that is locked
in. Arbitrage trading will continue until prices come into line (meaning: into line closely
enough that there is no more arbitrage profit after transactions costs).
In our example, arbitrage will force
Call price - Put price = S - PV(100)
The general Put-Call parity relationship that must hold in equilibrium is
Call price - Put price
C
-
FINC-UB.0043 Futures and Options Spring 2018
P
=
=
Asset price - PV(Strike price)
S
-
PV ( X )
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
4
Put-Call Parity and Option Properties from Portfolio Dominance
How to Exploit Violations of Put-Call Parity
Example: S = 100, X = 100, 3 month Call = 5.00, 3 month Put = 3.50, r = 8.00%
What should we do?
1. Does put-call parity hold?
C - P = 1.50
S - PV(X) = 100 - 100 / 1.02 = 1.96
No. C - P < S - PV(X). The put is too expensive relative to the call.
2. Arbitrage trade: Buy what is cheap and sell what is expensive
⇒ (Buy the call, write the put) and (sell short the stock - lend at the riskless rate)
⇒ initial cost: (5 - 3.5 ) + (-100 + 98.04 ) = - 0.46, negative "cost" = net cash inflow
3. At expiration in T = 3 months:
ST < 100
100 ≤ ST
Call
0
exercise (buy stock, pay 100)
Put
exercised (buy stock, pay 100)
0
Stock
use stock to cover short sale
use stock to cover short sale
Riskless asset
riskless bonds pay off 100
riskless bonds pay off 100
Total
0
0
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
5
Put-Call Parity and Option Properties from Portfolio Dominance
How to Exploit Violations of Put-Call Parity
Example: S = 100, X = 100, 3 month Call = 4.00, 3 month Put = 1.50, r = 8.00%
What should we do?
1. Does put-call parity hold?
C - P = 2.50
S - PV(X) = 100 - 100 / 1.02 = 1.96
No. C - P > S - PV(X). The put is too cheap relative to the call.
2. Arbitrage trade: Buy what is cheap and sell what is expensive
⇒ (Write the call, buy the put) and (buy the stock - borrow PV(X) at the riskless rate)
⇒ initial cost: (-4 + 1.5 ) + (100 - 98.04 ) = - 0.54 (net cash inflow)
3. At expiration in T = 3 months:
ST < 100
100 ≤ ST
Call
0
exercised (deliver stock, get 100)
Put
exercise (deliver stock, get 100)
0
Stock
use stock to cover option exercise
use stock to cover option exercise
Riskless asset
repay loan = -100
repay loan = -100
Total
0
0
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
6
Put-Call Parity and Option Properties from Portfolio Dominance
Beyond Arbitrage: Portfolio Dominance
Put-call parity is a fundamental relationship.
•
•
It must hold for (European) options.
It shows up all over in option strategies and price relationships.
The style of proof is very powerful.
•
Two positions that are equivalent to one another no matter what happens, must always cost the
same.
The principle of "Portfolio Dominance" (no portfolio dominance, actually) is a generalization
of "no-arbitrage."
•
If two positions are such that their payoffs are not always equal, but the first is never less than
the second and can be greater under some circumstances, then the first position must cost more
than the second.
Portfolio dominance leads to more general properties of option prices than we can get from
option pricing models like Black-Scholes.
•
•
We use it to prove a variety of price relationships, which seem intuitively obvious, must in fact
hold for all options under all circumstances. The proof is similar to that for put-call parity.
We can also prove some option properties that aren't so obvious, such as the fact that an
American call on a non-dividend paying stock should be worth no more than a European call
with the same terms.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
7
Put-Call Parity and Option Properties from Portfolio Dominance
Example: Portfolio Dominance Proof That
A Lower Strike Price Makes A Call Option More Valuable
Consider the payoffs at expiration on two calls on the same stock, that are identical except
that they have different strike prices, with X1 < X2 (e.g., X1 = 95, X2 = 100)
Payoff at Expiration Date T
ST ≤ X1
X1 ≤ ST < X2
X2 ≤ ST
Call 1
0
S T - X1
S T - X1
Call 2
0
0
ST - X2
(Call 1 - Call 2)
0
S T - X1
(Positive)
X2 - X1
(Positive)
The payoffs can both be zero, or else Call 1 pays more than Call 2.
Therefore, Call 1 must always sell for more than Call 2: C(X1) ≥ C (X2)
(They can be equal, but only if the probability is zero that the stock price at expiration is
above X1, in which case they are both worthless).
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
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Put-Call Parity and Option Properties from Portfolio Dominance
Option Properties from Portfolio Dominance
Call value increases with
• Higher stock price
• Lower strike price
• Longer time to expiration
• Higher interest rate
• Higher volatility
• Lower dividend payout
Put value increases with
• Lower stock price
• Higher strike price
• ???? time to expiration
• Lower interest rate
• Higher volatility
• Higher dividend payout
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
9
Put-Call Parity and Option Properties from Portfolio Dominance
More Option Properties that Can Be Derived from Portfolio Dominance
•
No early exercise of American calls unless the asset makes cash payouts
(dividends, coupon interest, etc.)
• Dividend payout can make it rational to exercise an American call early
(just before the stock goes ex-dividend).
•
Options are “convex” functions of both the asset price and the strike price.
•
"An option on a portfolio is worth less than a portfolio of options."
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
10
Put-Call Parity and Option Properties from Portfolio Dominance
Using Put-Call Parity to Understand a Call Option's Value
The market price is $6.43 for a 95 strike 1 month call option on a stock whose
market price is 100. What do you get from buying the call today instead of the
stock?
We can use put-call parity to decompose this into three intuitive elements.
Rearranging the put-call parity equation gives
C
=
C
=
S - PV(X) + P
( S - X)
+
(X - PV(X))
+
P
 the right not to own 


 interest saved by not

 intrinsic 
 + 
 +  the underlying if its 
Call value = 
 value 
 paying 95 until expiration 
 price falls below 95 


6.43
95 strike call
=
5
+
intrinsic value
FINC-UB.0043 Futures and Options Spring 2018
.61
+
1 month interest on 95 at 8%
Part II: Option Pricing and Hedging; Advanced Topics
.83
95 strike put
©2018 Figlewski
11
Put-Call Parity: Bonus Tracks
Rearranging Put-Call Parity
S = Price of Underlying Asset
C = Price of Call on S with Strike Price X
P = Price of Put on S with Strike Price X
Put-Call Parity
"Synthetic Long"
C
–
P
Buy
Call
and
=
Write
Put
Buy on Margin
S
–
PV(X)
Buy
Asset
and
Borrow
Rearranging the Put-Call Parity equation produces a variety of new positions
Covered Call
S
–
C
Buy
Stock
and
Write
Call
FINC-UB.0043 Futures and Options Spring 2018
=
Cash-Secured Put
PV(X)
–
P
Hold
Cash
and
Part II: Option Pricing and Hedging; Advanced Topics
Write
Put
©2018 Figlewski
12
Put-Call Parity: Bonus Tracks
Rearranging Put-Call Parity, continued
Protective Put
S
+
P
Buy
Stock
S
Buy
Stock
and
=
Buy
Put
Buy
Call
"Conversion"
–
C
+
and
90/10 Strategy
C
+
PV(X)
Write
Call
and
P
Buy
Put
"Reverse Conversion"
–S
+
C
–
P
Short
Stock
and
Buy and
Call
FINC-UB.0043 Futures and Options Spring 2018
=
Write
Put
and
Hold
Cash
Riskless Investment
PV(X)
Hold
Cash
=
Riskless Borrowing
– PV(X)
Borrow
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
13
Option Pricing Models: The Binomial Model
Option Pricing Models
Option properties derived from portfolio dominance are very general, but not very precise.
Many people tried to find a pricing formula that could say exactly how many dollars a given
option is worth. But for years, no one succeeded.
Fischer Black and Myron Scholes, along with Robert Merton, did it in the early 1970s, using
a kind of mathematics borrowed from physics to model the behavior of stock prices.
The new theory was very hard to explain to students (or to finance professors). Just for
pedagogical purposes, in his 1977 Investments textbook, William Sharpe introduced the
Binomial model, a very simplified framework in which an option could be easily understood
and priced.
However, it turned out that the simple classroom model could be extended and developed
into an extremely powerful and practical valuation tool, with flexibility to solve some kinds
of problems, like pricing an American put, that the Black-Scholes model can not handle. It
has been successfully extended to Trinomial and more complicated lattices for even more
power and flexibility.
You have already seen the one-period Binomial. The next few slides review that material.
We then extend the Binomial, and look at an important property known as Risk Neutral
Valuation.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
14
Option Pricing Models: The Binomial Model
Theoretical Underpinning of Derivatives Pricing Models
There are two types of valuation models for securities in modern finance theory, Equilibrium
models and Arbitrage-based models. We have seen this already for futures.
•
•
The Expectations Model is an equilibrium model. So is the Capital Asset Pricing Model.
The Cost of Carry Model is an arbitrage-based model.
Equilibrium Models
•
•
•
•
•
The security price is determined by aggregate supply and demand in the market.
The model value for a security should include a risk premium
Investors value each security relative to the entire set of available instruments.
Prices are brought into line with theoretical values because investors search out
undervalued securities and bid up their prices. They sell off overvalued securities,
driving their prices down until they become cheap enough to be worth holding.
All securities must be in equilibrium for the model to apply.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
15
Option Pricing Models: The Binomial Model
Theoretical Underpinning of Derivatives Pricing Models, p.2
Arbitrage-Based Models
•
•
•
•
•
The price is determined by the possibility of arbitrage between the derivative
security and its underlying asset.
The derivative is only valued relative to its underlying (and the riskless interest
rate)
The model price does not include a risk premium. All investors should value the
security the same way regardless of how risk averse or risk tolerant they are.
Prices are brought into line by arbitrage. In theory, because mispricing creates the
opportunity for riskless arbitrage, it only takes a single energetic arbitrageur to
force the market price to the model value.
Because the model only says how the derivative should be priced relative to the
underlying, it doesn't matter whether the underlying or any other security is in
equilibrium with respect to other traded assets.
Examples: The Binomial Model, the Black-Scholes Model, and the Cost of Carry
Model for futures
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
16
Option Pricing Models: The Binomial Model
The Binomial Model in Symbols and in Numbers
S = asset price at t=0.
We want the value of the call C at t=0
uS = 150
Over the next period, the stock can go to
only two possible prices:
up
to uS
(u > 1), or
down to dS
(d < 1).
S = 100
dS = 50
Suppose S = 100, and it can go to either
150 or 50 next period. u = 1.5, d = .5.
A 100-strike call C that matures next period
will either pay off 50, if the stock goes to
150, or 0, if the stock is at 50.
There is also a third asset we can trade: a
C = ??
riskless bond that costs 1 and pays a total
(principal plus interest) of R at t=1. You can
borrow money by selling the bond short.
Let R=1.1 in the example, i.e., 10% interest.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
Cu = 50
Cd = 0
©2018 Figlewski
17
Option Pricing Models: The Binomial Model
The trick is to construct a position using the stock and the bond that has the same payoff as
the call option in both the "up state" and the "down state." Such a position is called a
"replicating portfolio."
Buy h units of the asset and invest B dollars in bonds (negative B means borrowing). Choose
h and B to make the portfolio values in the up and down states be Cu and Cd.
h uS + R B
=
Cu
( h ( 150 ) + 1.1 B = 50 )
Down state : h dS + R B
=
Cd
( h ( 50 )
Up state :
+ 1.1 B =
0 )
Solving the two equations in two unknowns gives
h
=
B =
C u − Cd
uS − dS
1
R
=
50
150 − 50

 uS
dS
C
C
−
u
 uS − dS d
uS − dS


=
0.5
= − 22.73
Since they have the same payoffs, to avoid a riskless arbitrage, the call option and the
"replicating portfolio" must have the same price.
C = hS + B
FINC-UB.0043 Futures and Options Spring 2018
= 0.5 ( 100 ) − 22.73 = 27.27
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
18
Option Pricing Models: The Binomial Model
Substituting for h and B and simplifying leads to
C =
1
R
R − d

u−R
+
C
C
d
u −d u
−
u
d


The valuation formula can be simplified further if we define a new variable
p =
R −d
u−d
The final valuation equation is
C =
1
[ p C u + ( 1 − p ) Cd
R
]
This is called the "backward recursion" equation because it shows how to "roll back
through the tree" from the end to the start. The same equation is used in every step to price
the option at an earlier node when the values at the two nodes it branches to are known.
"Backward" means you begin the valuation process at expiration and move backward
through the tree to the starting date; "recursion" means you use the same formula over and
over again at every step.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
19
Option Pricing Models: The Binomial Model
Plugging in u = 1.5, d = 0.5, R = 1.10, Cu = 50 and Cd = 0, from our example gives
p =
R −d
u−d
1.10 − 0.5
= 0.6
1.5 − 0.5
=
which leads to an option value of
C =
1
[ p C u + ( 1 − p ) Cd
R
]
=
1
[ 0.6 × 50 + ( 1 − 0.6 ) × 0
1.10
]
C = 27.27
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
20
Option Pricing Models: The Binomial Model
Delta in the Binomial Model
The delta of an option is the number of units of the underlying one would use to create a
riskless hedge of the option.
Delta in the binomial is simply h, the number of shares in the replicating portfolio.
In our example, this is
h
FINC-UB.0043 Futures and Options Spring 2018
C u − Cd
uS − dS
h
=
=
50 − 0
150 − 50
= 0.5
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
21
Option Pricing Models: The Binomial Model
Delta in the Binomial Model, p.2
Let's check that a delta hedge of buying the option and shorting 0.5 shares does produce a
riskless position in our example.
The initial cost of the position is C – h S = 27.27 – .5 × 100 = – 22.73
There is a cash inflow from the market at the beginning.
Payoff
C u − h uS = 50 − 0.5 × 150 = − 25
Up state :
= 
0 − 0.5 × 50 = − 25
 Down state : Cd − h dS =
The position has the same value in both states, so it is riskless. Whether the stock goes up or
down, we have to pay out 25 at maturity.
What this trade amounts to is riskless borrowing of 22.73 from the market at t = 0, with
repayment (including 10% interest), of 25 at t=1.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
22
Option Pricing Models: The Binomial Model
Two Important Properties of the Binomial Model
1. The backward recursion valuation method works essentially the same way for any
contingent claim within the binomial framework. There is nothing in the way the pricing
equation was derived that limits it to calls.
2. The valuation equation does not involve the actual probabilities attached to the up and
down branches. The option value is the same for all probabilities. (The replicating
portfolio matches the option value in both possible states, so it doesn't matter how likely
each one is. The "probabilities" in the formula are artificial numbers that come out of the
solution to the "two equations in two unknowns" problem.)
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
23
Risk Neutral Valuation
Recall the Cost of Carry model for pricing gold futures. If S is the spot price for gold, say
$1600 per ounce, and r is the riskless interest rate, say 10%, what should the futures price be for
a 1 year futures contract?
F1 year = S ( 1 + r T/365) = 1600 x 1.10 = 1760
Suppose investors are so risk averse that they only hold gold if they expect the price to go up
25% a year. For them to pay $1600 an ounce for gold today, investors must be expecting it to
go up to 1600 x 1.25 = $2000 an ounce next year. What is the equilibrium gold futures price in
that case? It is the same 1760! The gold futures market obeys the cost of carry model because
you can buy gold and hedge away all the risk with futures, leaving a riskless position that must
earn the riskless interest rate, no matter how risk averse investors are. (Where does the risk go?)
Suppose investors are completely indifferent to risk: They are "risk neutral." Then any asset,
risky or not, should be priced to earn the same expected return. If gold is at 1600 in the spot
market when investors are risk-neutral, it is because the market expects the price to go up to
1600 x 1.10 = $1760 next year. What is the futures price? Still 1760, of course. But now, the
cost of carry model and the expectations model give the same answer.
"Risk neutral valuation" means the derivatives price in an arbitrage-based model is the
same as what it would be in the expectations model in a world of risk neutral investors.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
24
Risk Neutral Valuation
Risk Neutral Valuation for Options
Because of arbitrage, the option value in the Binomial Model is the same under all
probabilities. But it's often easier to solve a valuation problem if you know the probabilities.
Risk neutral valuation means that we can pick any convenient probabilities to solve the
valuation problem, and the answer we get must be the value under all possible probabilities.
An especially useful assumption is that investors are indifferent to risk and care only about
an asset's expected return. In this "risk neutral" world, all assets, including stocks and
options, will be priced to have the same expected return as a risk free asset.
In our Binomial world if p is the probability of an up step, then
E [ S1 ] = [ p u S + (1 − p) d S ]
And in a risk neutral world, the current price S is just the expected value of the next
period stock price S1 discounted by (1 + the riskless rate of interest).
S
=
E [ S1 ] / R
Combining the two, we have that the stock price in the Binomial follow the same equation that
the option does.
1
S =
FINC-UB.0043 Futures and Options Spring 2018
R
[puS
+ (1− p ) d S ]
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
25
Risk Neutral Valuation
Risk Neutral Valuation
Let's solve for the value of the "risk neutral probability" p using the stock prices in the
market.
In the example: S = 100, R = 1.1, u = 1.5, d = 0.5
SR
=
[ p uS
+ ( 1 − p ) dS ]
100 ( 1.10 ) = p ( 150 ) + ( 1 − p ) ( 50 )
110 = 50 + p ( 150 − 50 )
p =
110 − 50
150 − 50
=
60
100
= 0.6
In general: Solving the general equation for the risk neutral probability p gives
p =
FINC-UB.0043 Futures and Options Spring 2018
R −d
u−d
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
26
Risk Neutral Valuation
Risk Neutral Valuation for the Option
The current option value is the expected value of next period's price discounted at the
riskless rate. p is the risk neutral probability of an up step, so
C =
1
[ p C u + ( 1 − p ) Cd
R
]
This is exactly the form of the general valuation function we derived in solving the option
replication problem using arbitrage arguments!
C
1
[ 0.6 × 50 + ( 1 − 0.6 ) × 0 ]
1.10
30
=
= 27.27
1.10
=
Risk neutral valuation is a general property of all derivatives pricing models that are based
on arbitrage, like the Cost of Carry model for futures. It is an extremely useful and powerful
tool for obtaining valuation equations.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
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Extending the Binomial Model
To turn the Binomial into a practical valuation tool, it must allow more realistic asset price
behavior.
This is done by building a tree out of multiple binomial steps The time period of each one
can be made as short as one likes, so that the asset price can be made to go to as many
possible values as one likes in any given length of time, simply by subdividing the interval.
Price movements are multiplicative in the Binomial model, so an up move followed by a
down yields duS , which is the same price as a down followed by an up, udS. This means
that rather than having the number of nodes going up by 2, 22, 23, 24, ... as each new time
period is added, the lattice recombines, and the number of nodes goes up as 2, 3, 4, ...,
which keeps the procedure computationally manageable as the number of time steps grows.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
28
Extending the Binomial Model
Two Period Binomial Model
uuS
uS
udS
S
dS
ddS
Cuu
Cu
Cud
C
Cd
Cdd
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
29
Extending the Binomial Model
Pricing a Two-Period Call: S=100, X=100, u=1.5, d=.5, R=1.1
225
150
75
100
50
Cu = (.6 × 125 + .4 × 0)/1.1
= 68.18
25
125
C = (.6 × 68.18 + .4 × 0)/1.1
0
= 37.19
0
0
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
30
Extending the Binomial Model
Pricing a Put: Use the same parameters to price a 100-strike European put option.
225
150
75
100
50
Cu = (.6 × 0 + .4 × 25)/1.1
= 9.09
25
0
Cput = (.6 × 9.09 + .4 × 40.91)/1.1
25
= 19.83
Cd = (.6 × 25 + .4 × 75)/1.1
= 40.91
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
75
©2018 Figlewski
31
Option Pricing Using Risk Neutral Valuation in the Binomial Model
To illustrate how useful the principle of risk neutral valuation can be, let's see how the two
period European call and put would be priced.
There are 3 possible states after 2 periods:
uu (2 up moves):
probability = prob(uu) = p × p = 0.6 × 0.6 = 0.36
ud (1 up and 1 down): probability = prob(up then down) + prob(down then up)
= p (1 - p) + (1 - p) p = 2 p (1 - p) = 2 × 0.24 = 0.48
dd (2 down moves): prob(dd)
2-period discount rate:
= (1 - p) × (1 - p) = 0.4 × 0.4 = 0.16
R2 = 1.1 × 1.1 = 1.21
Risk Neutral Prices:
Call:
(1 / 1.21) × (0.36 × 125 + 0.48 × 0 + 0.16× 0 )
= 0.36 × 125 / 1.21
= 45 / 1.21 = 37.19
Put:
(1 / 1.21) × (0.36 × 0 + 0.48 × 25 + 0.16 × 75 ) = (12 + 12) / 1.21
= 24 / 1.21 = 19.83
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
32
Using the Binomial Model to Value American Puts
Early Exercise of American Puts
We have seen that a European put can have a fair value below its intrinsic value. In that
case, you would like to exercise it early, but you can't. This situation can't happen with an
American put. There is an "early exercise boundary" on the asset price, such that if the price
falls below the boundary, an American put should be immediately exercised.
Asset price
S0
T
time
X
exercise put here
Early exercise boundary
FINC-UB.0043 Futures and Options Spring 2018
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©2018 Figlewski
33
Using the Binomial Model to Value American Puts
Early Exercise of American Puts
It is rational to exercise an American put early. Time value has two components, one due to
"optionality" and the other due to the interest that can be earned on the exercise price. These
are both positive for a call option, so you won't exercise early. Optionality is also positive
for a put, but the interest effect is negative: If you are going to exercise the option and
receive the exercise price in cash, the longer you have to wait, the smaller is the present
value of that cash. Time value goes to zero for a deep in the money put, at which point it is
better to exercise it than to hold it any longer.
An American put is therefore worth more than a European put, because early exercise has
economic value. But valuing the early exercise feature of an American put is
mathematically tricky, so that there is no "simple" closed-form formula for the American put
price.
Put exercise is more likely with
• high intrinsic value, X - S
• short time to maturity
• low volatility
• high interest rate
• high dividend yield (payment of a dividend tends to delay exercise until after exdividend date)
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
34
Using the Binomial Model to Value American Puts
One of the great advantages of the Binomial is that it can be modified easily to deal with
early exercise and other features that are hard to handle in the Black-Scholes framework.
An American put should be exercised at any time prior to expiration if the intrinsic value
that would be received immediately is greater than the option value based on holding it over
the next period.
This is easy to build into the Binomial model. Since the rational option holder will make the
exercise choice that gives the larger value, simply compare the two values at each node and
put the larger one into the tree.
The backward recursion formula becomes
C American put
FINC-UB.0043 Futures and Options Spring 2018
= Max [
1
R
( p Cu
+ ( 1 − p ) Cd ) , X − S ]
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
35
Using the Binomial Model to Value American Puts
Pricing an American Put: Use the same parameters to price a 100-strike American put
option.
225
150
75
100
50
Cu = Max[(.6 × 0 + .4 × 25)/1.1 , 100 - 150]
= Max[ 9.09 , -50] = 9.09
25
0
Cput = (.6 × 9.09 + .4 × 50)/1.1
25
= 23.14
Cd = Max[(.6 × 25 + .4 × 75)/1.1 , 100 - 50]
= Max[ 40.91 , 50] = 50 (exercise early)
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
75
©2018 Figlewski
36
Using the Binomial Model
Incorporating a Dividend Payout in the Binomial Model
To price options on actual securities, the model must be adjusted to allow the underlying to
pay dividends or some other type of cash payout over time.
For a constant proportional dividend yield q, this is easily done by adjusting the risk neutral
probability p.
R is 1 plus the riskless interest rate per time step. Incorporate a proportional dividend yield
by dividing R by 1 plus the dividend rate per time step. Set
p =
R /(1 + q ) − d
u−d
Examples:
Stock index portfolio:
Foreign currency:
Futures
FINC-UB.0043 Futures and Options Spring 2018
q = percent dividend yield
q = foreign riskless interest rate
q = riskless interest rate (set R/(1+q) to 1)
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
37
Using the Binomial Model
Setting Binomial Model Parameters Based on Market Data
To use the model for pricing real-world options, we must set its parameters to
match market values.
•
Initial stock price, S, strike price X, and time to maturity T are already known.
•
Let ∆t be the length of one time step. In an N-step tree, ∆t = T / N .
•
If r is the (continuously compounded!) annual interest rate and y is the annual
continuously compounded proportional dividend yield, set R = e r ∆t and
(1+q) = e y ∆t .
•
Volatility per step v = σ ∆t where σ is the annualized volatility. This must be
turned into values for u and d. There are several common ways to do this.
° Easiest: Set u = e v
and d = e − v
° Fastest convergence:
Set
u = e
r ∆t − v 2 / 2 + v
and d = e
r ∆t − v 2 / 2 − v
This builds the riskless interest rate into the tree as the mean return on the stock.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
38
Option Pricing Models: Black-Scholes
The Binomial model allows us to approximate option values to as close a degree of accuracy
as we want. It is also much more flexible than Black-Scholes, so it can be used to solve
many valuation problems, such as pricing an American put, that the Black-Scholes model
can't handle.
However, the Binomial is just an approximation, not a "closed-form" solution. A closed
form solution is an equation that gives the exact option value as a function of a set of input
parameters.
Also, the Binomial is less efficient than a closed-form equation. It may take a long time to
converge to the exact option value as the number of time steps in the tree is increased, and
convergence is not monotonic (for example, a 200 step tree may give a less accurate answer
than a 100 step tree).
This session introduces the Black-Scholes model. This classic model is a closed-form
solution to the option pricing problem. It is universally used in practice (although not in the
way the theory says it should be used) and it points the way towards similar valuation
equations for other types of derivative instruments with option features.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
39
Option Pricing Models: Black-Scholes
Convergence to the True Option Value in the Binomial Model
Note the strong even-odd pattern and also that convergence isn't uniform: you can be closer
to the right answer with 50 steps than with 80 here.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
40
Option Pricing Models: Black-Scholes
Dynamic Hedging to Replicate Option Payoffs
The Binomial Model set up a framework in which the underlying asset and the riskless bond
could be combined to create a position that exactly replicates the payoff on the option.
The Black-Scholes model is derived in a similar way: The option and the stock are
combined to create a hedged position that is like a riskless bond.
Like the bond, the riskless option-stock hedged position must return the riskless rate of
interest. This leads to a fair price for the option.
Black and Scholes' assumptions permit a much more realistic price process than the
Binomial for the underlying asset, while still allowing a riskless hedge to be constructed
over the option's entire lifetime. However, this requires a dynamic hedging strategy,
because the position is only riskless over the next instant in time, and then it must be
rebalanced.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
41
The Standard Model for Asset Returns
Price changes for securities like stocks have several stylized features that need to be
incorporated into whatever model is used for the "returns process:"
• the price is observable (more or less) continuously
• random fluctuations occur all the time
• prices follow a "random walk," meaning that the random fluctuations are independent from one
•
period to the next, even at the shortest intervals
the distribution of percentage returns is (approximately) normal
These properties are expressed formally in the form of a "lognormal diffusion process."
This is the standard assumption for derivatives modeling, particularly options.
We will look at the lognormal diffusion model more closely later in the course, when we get
to options. For now, we will just assume some of the properties of the model hold, without
getting into details.
Key assumptions: Over a period of time of length T,
• the (continuously compounded) return follows a normal distribution
• security prices follow a lognormal distribution (the logarithm of price is normally distributed)
• The "Square Root of T Rule" for volatility:
° the variance of the return is proportional to the length of the time interval
VAR[R] = VAR[ln(ST/S0)] = σ2T; standard deviation of return = σ√T
• returns measured over non-overlapping time periods are statistically independent
FUNC-UB.0043 Futures and Options Spring 2017
Part I: Forwards and Futures
©2017 Figlewski
42
The Standard Model for Asset Returns
Standard Normal Probability
0.45
0.4
About 2/3 of the probability falls
within plus or minus 1 standard
deviation of the mean.
0.35
Probability
0.3
0.25
0.2
0.15
0.1
0.05
0
-4
-3
-2
-1
0
1
2
3
4
Standard deviations
FUNC-UB.0043 Futures and Options Spring 2017
Part I: Forwards and Futures
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43
The Standard Model for Asset Returns
Lognormal Probability
0.45
0.4
Probability of stock price in 1 year
Initial price = 100
Mean return = 6%
Volatility = 25%
0.35
Probability
0.3
Expected value of price in
t = one year
= V0 e(r + σ^2/2)t
= 100 e(.06 + .25^2/2)
= 109.55
0.25
0.2
0.15
0.1
0.05
0
25
50
75
100
125
150
175
200
225
250
x value
FUNC-UB.0043 Futures and Options Spring 2017
Part I: Forwards and Futures
©2017 Figlewski
44
The Standard Model for Asset Returns
The Asset Price Process
The Black-Scholes model assumes the price of the underlying asset, S, follows a "lognormal
diffusion" process:
dS = µ S dt + σ S dz
where
dS = the change in stock price over the next instant
µ = the "drift," that is, the average rate of capital gains as a
continuously compounded annualized figure
dt = an "instant"
σ = the volatility, expressed as an annual rate
dz = "Brownian motion," a very small random shock to the
price over the next instant.
dz has mean zero and variance 1 dt. The standard deviation of dz is 1 dt (which is a lot
bigger than dt when dt is very small. This means the volatility term dominates the drift term
over short time periods.)
Current dz is independent of all dz in previous (and future) periods
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
45
The Standard Model for Asset Returns
The Asset Price Process, p.2
The lognormal diffusion model of stock price changes has several important
implications:
•
Price paths are continuous. It is impossible for the price to jump from one level to
another without passing through every price in between.
•
Prices fluctuate randomly at all points in time, but the randomness is independent
from period to period. The price behavior is often called a "random walk." (This is
intuitive but imprecise mathematically. The exact description is a "semimartingale": price changes are random and independent, but there can be a
nonrandom drift and variance can change from period to period. )
•
Given a starting value S0 , the log price change over the period from 0 to T is
given by R = ln ( ST / S0 ).
•
R has a normal distribution, with: mean = µ T
•
and standard deviation = σ
ST has a lognormal distribution with expected value = S0 e
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
T
( µ + σ2 / 2 ) T
©2018 Figlewski
46
Option Pricing Models: Black-Scholes
Underlying Assumptions of the Black-Scholes Option Pricing Model
•
Options are European
•
"Perfect" markets -- no transactions costs, no taxes, no constraints on short selling
with full use of the proceeds, no indivisibilities, etc. (doesn't say anything about
"efficient" markets)
•
No limits on borrowing or lending at a known risk free rate of interest
•
The price of the underlying asset follows a "lognormal diffusion" process
•
The return volatility of the underlying asset is known
•
No dividends or cash payouts from the underlying asset prior to option maturity
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
47
Option Pricing Models: Black-Scholes
The Partial Derivatives of the Call Value Function
The call valuation model derived by Black and Scholes gives the option price as a function
of 5 variables (6, if one allows dividends, which the original BS model did not).
Call = C( S, X, T, r, σ )
where,
S = stock price
X = strike price
T = time to option expiration
r = riskless interest
σ = volatility
The sensitivity of the option value with respect to a change in one of the parameters is given
by the relevant partial derivative.
We normally think of the stock price and time as changing, while the other parameters are
fixed. But it can be useful to know how much the option price would be affected by a
slightly different volatility parameter or riskless interest rate, so those partial derivatives are
of interest as well.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
48
Option Pricing Models: Black-Scholes
The Partial Derivatives of the Call Value Function
A partial derivative is like an ordinary derivative in calculus. If one is interested in a
function f(x), defined for a single variable x, the derivative of f(x) with respect to x tells
how much the function value will change per unit change in x.
We write the derivative as
df
dx
For a function of more than one variable, the same concept applies. The partial derivative
with respect to one of the variables in the function tells how much the function will change
per unit change in the variable in question, holding all of the other variables in the function
constant. If f(x,y) is a function of two variables x and y, we write the partial derivatives as
∂f
∂x
and
∂f
∂y
Example: Consider the function f = 2 x2 y :
∂f
∂f
∂ 2f
∂ 2f
∂ 2f
2
= 4=
xy ;
2 x=
;
4=
y;
4=
x;
0
2
2
∂x
∂y
∂xy
∂x
∂y
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
49
Option Pricing Models: Black-Scholes
The Fundamental Partial Differential Equation of Derivatives Pricing
The Black-Scholes model equation is the solution to a partial differential equation (PDE).
Such an equation exists for every derivative, in many cases differing only in the boundary
conditions it must satisfy. The PDE can be used for derivative pricing with numerical
approximation techniques even when there is no closed form solution.
Here is a quick look at the fundamental PDE for the Black-Scholes model and how it leads
to the Black-Scholes option pricing equation. (The next few slides are provided exclusively
for your viewing pleasure. This material will not appear on any homework or exam.)
The price S of the underlying asset is assumed to follow the lognormal diffusion process
dS
where
=
µS dt
+
σS dz
µ is the instantaneous mean return
σ is the volatility of the return.
dz is standard Brownian motion
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
50
Option Pricing Models: Black-Scholes
Ito's Lemma
The mathematical tool for dealing with diffusion processes is Ito's Lemma.
We are interested in a variable C that is a function of S and time: C = C(S,t).
Ito's Lemma gives the equation for the diffusion process followed by C.
Ito's Lemma
∂C
∂C
1 ∂ 2C
2
dC =
dS +
dt +
(dS)
∂S
∂t
2 ∂S2
The term (dS)2 in this expression is evaluated using the 3 multiplication rules:
dt dz = 0;
(dt)2 = 0;
(dz)2 = dt
which come from the fact that as dt goes to zero, any higher power of dt, like (dt)2 and (dt)3/2
becomes infinitesimally small relative to dt and drops out of the equation.
(For example, suppose dt is 1/1000th of a year (about 1/3 of a day). (dt)2 would be
1/1,000,000th of a year: a thousand times smaller.)
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
51
Option Pricing Models: Black-Scholes
Applying Ito's Lemma to the option value equation C(S,t) gives
∂C
∂C
1 ∂ 2C
2
dC =
dS +
dt +
(dS)
∂S
∂t
2 ∂S2
=
𝜕𝜕𝜕
𝜕𝜕𝜕
µ S dt +
𝜕𝜕𝜕
𝜕𝜕𝜕
σ S dz +
𝜕𝜕𝜕
dt
𝜕𝜕t
+
1 𝜕𝜕2 C
2 𝜕𝜕S2
σ2 S2 dt
Notice that the first term is the option's "delta" times the change in the stock price.
The last term comes from using the multiplication rules from the previous page in
calculating (dS)2.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
52
Option Pricing Models: Black-Scholes
The Fundamental Partial Differential Equation of Derivatives Pricing
The value of any derivative instrument C(S,t), such as a call option, only varies with the
price of the underlying asset and time. It can be hedged over the next instant by a short
position in the underlying asset.
Hedge: Sell delta = ∂C / ∂S units of the asset for each call.
The hedged position is worth V = C - (∂C/ ∂S) S. The dynamics of the hedge portfolio are
given by
∂C
∂C
1 ∂ 2C 2 2
σ S dt
dV =dC −
dS = dt +
∂S
∂t
2 ∂S2
Notice that both the dz term and the term involving the stock's mean return µ have canceled
out. Because the position is perfectly hedged, there is no risk and no risk premium in this
expression. (Sounds a lot like risk neutral valuation: Risk aversion doesn't enter the
valuation equation.)
The position is perfectly hedged over the next instant dt, so it must earn the riskless interest
rate. Otherwise there would be an arbitrage. Therefore, we also have
=
dV
FINC-UB.0043 Futures and Options Spring 2018
rV
=
dt
r (C −
∂C
S) dt
∂S
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
53
Option Pricing Models: Black-Scholes
The Fundamental Partial Differential Equation of Derivatives Pricing
Combining the two relationships that dV must satisfy leads to the following partial
differential equation (PDE), that must hold for every derivative security:
The Fundamental Partial Differential Equation of Derivatives Pricing
rC
-
∂C
rS
∂S
-
∂C
∂t
-
σ2 2 ∂ 2 C
S
2
∂ S2
=
0
The solution to a partial differential equation is a function. In this case, that function is the
Black-Scholes valuation model for the derivative security C.
This same equation holds for a call and a put both. What makes the formula price one rather
than the other are 3 boundary conditions.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
54
Option Pricing Models: Black-Scholes
The Fundamental Partial Differential Equation of Derivatives Pricing, cont.
To solve this PDE for a particular derivative security, one must add boundary conditions,
that specify what happens to the value at maturity date T, what happens at some date t < T
before maturity if S goes to zero, and what happens when S grows infinitely large.
Boundary conditions for a European Call Option:
if the stock goes to 0:
C ( S, T) = Max ( ST − X, 0 )
limit C ( S, t ) = 0
if the stock goes very high:
limit
at option maturity:
S→0
∂ C ( S, t )
∂S
S→∞
= 1
Boundary conditions for a European Put Option:
if the stock goes to 0:
Max ( X − ST , 0 )
limit P ( S, t ) = PV (X)
if the stock goes very high:
limit P ( S, t )
at option maturity:
FINC-UB.0043 Futures and Options Spring 2018
P ( S, T )
=
S→0
S→∞
= 0
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
55
Option Pricing Models: Black-Scholes
Fundamental PDE for Call Option
30
25
20
Option Value 15
25-30
20-25
15-20
10
10-15
5-10
0-5
Time to expiration
5
0.70
0.35
120
118
116
114
112
0.00
110
108
106
104
102
100
98
96
94
92
90
88
86
84
82
80
0
Stock Price
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
56
Option Pricing Models: Black-Scholes
The Black-Scholes Model
Let
S = asset price
X = strike price
r = riskless rate
T = maturity
σ = volatility
Assume the option is European and the underlying asset pays no dividends.
Call option value:
C = S N[ d
]
[
− X e − rT N d − σ T
]
2

σ
T
ln S / X +  r +

2 

where d =
σ T
Call delta:
δCALL = N [ d ]
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
57
Option Pricing Models: Black-Scholes
The Black-Scholes Model
Under the same assumptions as above, the Black-Scholes model value for the
European put can be derived in the same way, but it is obtained more easily
directly from put-call parity.
Put option value:
[
]
P = X e − rT N − d + σ T − S N[ − d
Put delta:
FINC-UB.0043 Futures and Options Spring 2018
δPUT =
]
− N[ − d ]
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
58
Volatility
Modern option pricing models following Black-Scholes all have a major role for the
volatility of the underlying asset's return. This is the one parameter in Black-Scholes that is
not observable, so naturally there is a lot of dispersion around the average volatility
expectation in the market.
In Black-Scholes, volatility is assumed to be a constant known parameter. But in the real
world it is neither constant nor known. This session will review what we know about
volatility, how it goes into option prices, and how the wide variety of volatility-based
products can be used to speculate or hedge volatility.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
59
"Stylized Facts" about Volatility
Here are some common findings from research on the behavior of asset volatility
•
•
•
volatility is not constant; it changes substantially over time
•
in equity markets, volatility increases when stock prices fall, and (may) decrease
when prices rise (this is often called the "leverage effect")
•
implied volatility has a regular structure across options with different strike prices,
known as the "smile" or the "skew"
•
implied volatility also shows systematic "term structure" effects for options with
different maturities
periods of high volatility and periods of low volatility cluster together
there appears to be "mean reversion" in volatility; periods of unusually high or low
volatility tend to be followed by a reversion to more normal behavior
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
60
Estimating Realized Volatility
Three Techniques for Estimating / Forecasting Volatility from Historical Data
1. Historical volatility:
• Compute K log returns from past prices: Rt-k = ln( St-k / St-k-1 ), for k = 1,...,K
•
Volatility estimate = annualized standard deviation of {R}
date
stock
t
t-1
t-2
t-3
t-4
102
101
97
99
100
log return return squared
0.985%
4.041%
-2.041%
-1.005%
average 0.495%
annualized (x 255) 126.24%
volatility
FINC-UB.0043 Futures and Options Spring 2018
9.70677E-05
0.001632931
0.000416522
0.000101009
0.00056188
0.14328003
37.85%
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
61
Estimating Realized Volatility
Three Techniques for Estimating / Forecasting Volatility from Historical Data
2. Exponentially weighted moving average
•
•
Compute log returns from past data as above
Downweight data as it ages, by multiplying each squared deviation by wk, for some
weight w < 1.0. (Riskmetrics uses w = 0.94.)
•
Volatility estimate = annualized value of
k = max
∑ w
k =0
date
stock
t
t-1
t-2
t-3
t-4
102
101
97
99
100
average
annualized (x 255)
log return return squared
0.985%
4.041%
-2.041%
-1.005%
0.495%
126.24%
9.70677E-05
0.001632931
0.000416522
0.000101009
k
R 2t − k
weight
factor = .90
1
0.9
0.81
0.729
∑ wk
k =0
wgt x ret sq
9.70677E-05
0.001469638
0.000337383
7.36357E-05
sum
3.439 0.001977724
weighted average 0.000575087
annual variance (x 255) 0.146647175
volatility
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Estimating Realized Volatility
3. GARCH (Generalized Autoregressive Conditional Heteroskedasticity)
Model variance at date t as a combination of

last period's variance, σ2t-1

last period's squared random price shock ε2t-1
The simplest GARCH model has two equations:
= µ + εt ,
εt is distributed as Normal (0,σ2t)
Return equation:
rt
Variance equation:
σ2t = C + a σ2t-1 + b ε2t-1
(Typical values are 0.90 to 0.95 for a and 0.05 to 0.08 for b. The sum should be < 1.0.)
For stocks an asymmetry term is added to make volatility go up when the stock price falls.
The GARCH model for stock returns: σ2t = C + a σ2t-1 + b ε2t-1 + d ε2t-1 (if εt-1 < 0)
Note that in the GARCH framework, time is not continuous.
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Estimating Realized Volatility
Practical Issues in Estimating Volatility from Past Prices
If the price of the underlying asset did follow a lognormal diffusion with constant mean and
volatility, as assumed by Black and Scholes, you would get the best volatility estimate by
using as much past data, sampled at as fine an interval, as you could get. But with real
world prices, several practical issues arise:
•
what observation interval to use (daily? monthly? intraday?)
• suggestion: higher frequency is better, so use daily data; intraday returns require special handling
•
whether to estimate the mean
• NO!
•
how much past data to include
• as much as possible, but not from much different economic environments
•
how to deal with "outliers," i.e., events like October 19, 1987
• there is no perfect answer; use judgment and consider how sensitive the results are to the outliers
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Estimating Realized Volatility
Conclusions on Forecasting Volatility: Historical Data
Here are several general conclusions I have reached based on my research on volatility
prediction
•
•
•
•
•
•
•
•
Different methods should be compared in terms of out of sample forecasting
performance
Accuracy of all methods is low
Using data sampled at very short intervals (daily or less) requires careful adjustment
for "noise" arising from the trading process (e.g., "bid-ask bounce")
It is better to assume the mean is 0 than to take deviations around the sample mean.
Simpler models, such as straightforward use of measured historical volatility over a
long sample period seem to be about as accurate as more complicated models, and
are more robust.
Volatility forecasts for long horizons seem to be more accurate than for short
horizons
GARCH models appear to work well over very short horizons, but only if there is a
lot of data available to estimate model parameters
GARCH seems to work best for equities.
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Implied Volatility
Implied volatility is the value of the volatility input to an option pricing model that makes
the model value equal the option price observed in the market.
IV is the solution to
C(S, X, T, r, IV) = Cmarket
(The actual value for IV must be found by a search process.)
IV should impound "the market's" forecast of volatility. IV is felt by many to be the best
volatility estimate possible, because the market has access to much more information than
any model can incorporate.
There is a one-to-one correspondence between implied volatility and option price. In some
markets, options are quoted in terms of implied volatility, rather than price. OTC foreign
currency options are an important example.
Note that implied volatility depends on the option pricing model used to calculate it. IV as
commonly reported is always computed from the Black-Scholes model (with dividend
correction, and sometimes with an adjustment for American exercise).
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Vega in Action
Implied volatility July 31
July 31Closing price
August 1Closing price
Implied volatility August1
67
Implied Volatility
The VIX Volatility Index
The CBOE computes, publishes, and now trades futures and options based on a composite
index known as the VIX. The VIX index is designed to measure the standard deviation of the
S&P500 stock index over the next month, as implied by the current market prices of SPX index
options. The index is an interpolated value for a one month horizon, extracted from the nearest
to expiration and the next nearest contracts.
The original formula for computing the VIX was changed in 2003. The "Old VIX," (still
computed and published as the VXO), was based on 8 at-the-money calls and puts on the OEX
index and used the Black-Scholes model to extract the implied volatilities. These IVs were
combined into a weighted average 30-day at-the-money implied volatility. (There was also a
technical problem in annualizing the index, which made it significantly biased upward.)
The new VIX uses all out of the money SPX index calls and puts expiring just before and just
after 30 days and extracts, not individual IVs, but the whole risk neutral probability distribution
(without using Black-Scholes or any other pricing model). The implied volatility is the
standard deviation from this implied distribution.
Futures contracts on the VIX began trading in 2004 and VIX options in 2006.
Although it is supposed to be an estimate of volatility, the VIX is widely thought of and
referred to, as "the market fear gauge".
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2500
90
Historical and Implied Volatility for S&P 500 Index
80
2000
70
1500
50
40
1000
30
20
500
10
SPX
VIX
Historical Volatility (last 6 months)
0
0
9/4/1990 5/31/1993 2/25/1996 11/21/1998 8/17/2001 5/13/2004 2/7/2007 11/3/2009 7/30/2012 4/26/2015
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Volatility
S&P 500 Index
60
The Volatility Surface
70
Volatility: Flowchart of How Volatility Gets into Option Prices and How it is Extracted
Historical
price data
Statistical Estimate
of
Future Returns
Forecasts and
other information
Investors'
Estimate of
Future Returns
Hedging
(risk aversion)
Demand for Options
Market
Option Prices
Invert the BlackScholes Model
Implied BS Volatilities
Volatility smile
What a financial
econometrician calculates
What the market actually
predicts: the "P" density
also known as "empirical" or
"real world" probabilities
What the options market embeds in
option prices, including risk premia:
the "Q" density,
Model-free
Volatility
calculation
"Risk-neutral" Probabilities
Volatility surface
Risk Neutral Density
The original
VIX Index
VXO
Model-free
Implied Volatility;
the VIX Index
71
Implied Volatility
The Real Mystery: Where Does the Volatility Smile Come From?
A Variety of Possible Explanations:
1. The underlying returns distribution is not Normal. ("Fat tailed" distributions)
• too many "big" returns (both positive and negative); a Student-t distribution with
about 7 degrees of freedom fits the data better
2. Volatility of a stock depends on the ratio of equity to debt in the firm's capital structure,
and that changes when the stock price moves. (The "leverage effect")
3. Volatility is stochastic
• GARCH (volatility is a function of the asset price change)
• Two-factor models (in which volatility is subject to random changes that are at
least partly independent of the price change)
4. Stock prices can make large jumps (non lognormal "jump-diffusion processes")
5. Investors are "Crash-o-phobic" and are willing to pay extra for the protection of out of
the money puts. In the money calls (that have strike prices below the current stock price,
like out of the money puts) must also have high implied volatilities or else they would
violate put-call parity.
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Trading Volatility: Variance Swaps and Vol Swaps
There are now multiple ways to trade volatility as an investible asset. The easiest is a
variance swap. This is a forward contract on the difference between the realized variance
of returns over some period ending at a future date and a strike level set at the beginning.
Example: A trader who expects high volatility in the stock market until the end of the year
2016 could "buy" a variance swap with a strike variance rate of (0.20)2 and maturity Dec. 31.
The quote is often done in terms of the volatility implied by the given variance, which is
20% in this case.
Realized variance is calculated as the annualized mean of Rt2 , where Rt is the log return, for
dates t from now through Dec. 31. The difference between realized variance and the strike
is multiplied by a notional principal to get the payoff in dollars.
A volatility swap is essentially the same thing, except the payoff is in terms of the
difference between realized volatility and the strike.
A volatility swap seems to make more sense, since volatility is what investors care about,
but valuation is trickier mathematically, meaning it is a lot harder to hedge than a variance
swap, so vol swaps are less popular.
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Trading Volatility: Derivatives Based on the VIX
In 2004, the CBOE introduced trading in futures contracts based on the VIX. Contract
size is $1000 times the VIX index, with minimum tick size of 0.01 (1 volatility basis point).
There is no cost of carry pricing model for VIX products, since you can't store the VIX.
In 2006, the CBOE added VIX options, i.e., options whose payoffs are equal to
$100 * Max(VIXT – X, 0), where T is option maturity and X is the strike level.
Volatility products have become immensely popular. There are now dozens of related
volatility contracts for different commodities and time horizons.
There are also numerous exchange traded funds that offer exposure to volatility (of stocks,
or oil, or gold, or...) just like a mutual fund. These trade like stocks, not futures or options.
But they all use VIX futures to create their payoffs.
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Trading Volatility: Derivatives Based on the VIX
Trading the VIX and financial products based on the VIX is very different from trading
stocks or commodities, because the VIX itself cannot be bought and held. It is not
investible. Exchange-traded VIX-based products like ETFs are all based on VIX futures.
Gold futures are tied to the current price of gold, because you can buy gold today, hedge by
selling futures, and lock in the return on the trade. Similarly, you can buy a portfolio of
stocks, sell short index futures, and carry that hedged position to futures maturity.
By contrast, the VIX is more like a temperature. The current level can be observed all the
time, but you can't buy the VIX, carry it over time and deliver it against a future or option
contract.
This means:
1. The VIX future is based on expectations about what the VIX will be at futures maturity.
There is no arbitrage-based pricing model for the VIX.
2. There is therefore no direct connection between what the spot VIX does and how any
VIX products behave, just as today's temperature tells us little about what the temperature
will be 30 days from now.
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Implied Volatility
The Information Content of Implied Volatility
Question: Is implied volatility from an option's market price an efficient forecast of future
volatility of the underlying asset?
•
Is IV an unbiased forecast?
•
Does IV impound all of the information contained in historical volatility?
This issue has been examined many times in the literature for different markets. Most
researchers find IV biased, but that it contains information, and generally more than
historical volatility.
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Implied Volatility
The Information Content of Implied Volatility
The idea that implied volatility is an efficient forecast of future realized volatility involves
a joint hypothesis.
1. The implied volatility IV is equal to the market's volatility forecast:
IV = EMKT[σ]
2. The market's forecast is rational
σ = EMKT[σ] + ε
where ε has mean 0 and minimum variance given the currently available information.
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Implied Volatility
The Information Content of Implied Volatility: Standard tests
1. "Rationality test regression:" If F is a rational forecast of volatility σ, then
σ =
F + ε
To test this, run the regression
σt = α + β Ft + ut
Test α = 0 and β = 1.0
2. "Encompassing regression:" With multiple forecasts F1 and F2, run this regression
σt = α + β1 F1t + β2 F2t + ut;
If the F1t forecast impounds all of the information contained in F2t, then α = 0.0, β1 = 1.0,
and β2 = 0
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Implied Volatility
The Information Content of Implied Volatility: Typical Test Results
1. "Rationality test regression:"
σt = α + β Ft + ut
Test results:
α ≅ 0.06 and
β ≅ 0.65
2. "Encompassing regression:" With multiple forecasts, run this regression
σt = α + β1 ImpliedF1t + β2 HistoricalF2t + ut;
Test results:
α ≅ 0.06 and β1 ≅ 0.65, and β2 ≅ 0
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Implied Volatility
Conclusions on Volatility Prediction: Implied Volatility
•
The volatility smile shows that the basic Black-Scholes option pricing model does
not fully explain how options are priced in the market.
•
Implied volatility nearly always contains information about the volatility that will
occur in the future, but it is biased as a forecast.
•
Even when IV is shown to contain a significant amount of information about future
realized volatility, if it is biased, it will not necessarily be an accurate forecast
unless the bias is corrected.
•
More sophisticated option pricing models can be constructed that are consistent
with the existence of a volatility smile, but are they the true explanation for it?
•
Despite all of these issues, market makers prefer to use implied volatility in their
models, because they want the model values to match the prices they are seeing in
the market. This leads to the use of "practitioner Black-Scholes," which is the
Black-Scholes equation, but with a different volatility input for each option.
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Delta Hedging and Beyond
The Partial Derivatives of the Call Value Function
The partial derivative of the call value with respect to a small change in the stock price is the
option's delta. Delta is often written using the Greek letter delta, either δ (lower case) or ∆
(upper case). We will use δ.
δ =
∂C
∂S
Delta serves the same function as the hedge ratio in a futures hedge. It tells how many units
of the underlying asset one should trade in order to hedge the market risk exposure of the
option.
For example, if δ = 0.50 for a given call option, the position that is long one call and short
0.50 shares of stock will be hedged against a (small) change in the stock price up or down.
This is called a "delta hedge" and the hedged position is said to be "delta neutral."
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Delta Hedging and Beyond
The Partial Derivatives of the Call Value Function
A very important problem with delta hedging is that delta changes as the stock price moves.
The delta of a call option ranges from 0, for an option that is very far out of the money, to
1.0 for a call that is very deep in the money. Delta hedging requires rebalancing the
proportions of stock and the option continuously, as the stock price moves and as time
elapses.
How much delta changes as S moves is given by the partial derivative of delta with respect
to S. This is the second partial derivative of the option value with respect to S. This concept
is called by another Greek letter, gamma ( γ ). Gamma is related to the curvature of the
option value function. Positive gamma means the function is convex.
∂δ
∂2 C
γ =
=
∂S
∂ S2
The other partial derivatives of the option function are also of importance in hedging, so the
Greek alphabet is well represented. These measures of exposure to different types of risk
affecting option value are commonly known as "the Greeks."
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Delta Hedging and Beyond
"Greek Letter" Risk Exposures for Options
Delta (δ) - The change in the option value produced by a 1 point change in the price of the
underlying asset. Delta measures exposure to Market Risk.
Gamma (γ) - The change in the delta produced by a 1 point change in the price of the
underlying asset. Gamma measures Convexity, which turns into risk for a delta neutral
hedge.
Vega - The change in the option value produced by a 1 percentage point change in the
volatility of the underlying asset. Vega measures exposure to Volatility Risk (despite not
being a true letter in the Greek alphabet). Vega is sometimes written as Λ (which is actually
upper case lambda) or as ν, (which is a Greek nu) .
Theta (θ) - The change in the option value produced by a 1 day drop in the time to maturity.
Theta measures Time Decay.
Rho (ρ) - The change in the option value produced by a 1 percentage point change in the
interest rate. Rho measures exposure to Interest Rate Risk.
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Delta Hedging and Beyond
Greek Letter Risk Exposures for Option Positions
Computing the total Greek letter risk exposures for a position containing several options is
straightforward.
In calculus, the derivative (or partial derivative) of the sum of two functions is the sum of
the (partial) derivatives of the individual functions. That is,
∂
( f ( x , y) + g ( x , y)
∂x
)
=
∂
f ( x , y) +
∂x
∂
g ( x , y)
∂x
For example, if our position consists of N1 options of type C1 (calls, puts, or any other
kind of derivative) and N2 options of type C2, the delta of the combined position is just:
Combined delta = N1 δ1 + N2 δ2
and the same for all of the other Greek (and near-Greek) letters.
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Delta Hedging and Beyond
In-Class Problems
Here are the current market prices for XYZ stock and two XYZ options. The Greek letter
risk exposures come from the Black-Scholes model. The interest rate is 8% and the implied
volatility is 0.25.
XYZ Stock
XYZ Call 105 strike, 1 month
XYZ Put 95 strike, 1 month
Market price
100
1.25
0.83
delta
1
0.29
-0.21
gamma
0
0.047
0.039
vega
0
.099
.084
theta
0
-.044
-.030
You are long the 105 call on 100,000 shares.
1. How would you set up a delta hedge for this position?
2. What would the overall hedged position be worth? (What is the net cost to set it up?)
3. What are the Greek letter exposures for the overall position?
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Delta Hedging and Beyond
In-Class Problems
XYZ Stock
XYZ Call 105 strike, 1 month
XYZ Put 95 strike, 1 month
Market price
100
1.25
0.83
delta
1
0.29
-0.21
gamma
0
0.047
0.039
vega
0
.099
.084
theta
0
-.044
-.030
1. Position delta is 100,000 x 0.29 = 29,000. Hedge by shorting 29,000 shares.
2. Cost to set up is negative:
Calls 100,000 x 1.25 = 125,000
Stock -29,000 x 100 = -2,900,000
Total
= -2,775,000
3.
100,000 x 0.29 + (-29,000) x 1 = 0
100,000 x 0.047 + (-29,000) x 0 = 4,700
100,000 x 0.099 + (-29,000) x 0 = 9,900
100,000 x -.044 + (-29,000) x 0 = -4,400
delta:
gamma:
vega:
theta:
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Delta Hedging and Beyond
In-Class Problems
Tomorrow, XYZ stock opens at 95. Here is the new set of option prices and Greek letters.
Market Price
XYZ Stock
XYZ Call 105 strike, 1 month
XYZ Put 95 strike, 1 month
95
0.30
3.35
delta
1.0
0.10
-0.46
gamma
0
0.025
0.044
vega
0
.047
.108
theta
0
-.021
-.052
4. If you liquidated right now, what would the profit or loss on the hedged position be?
5. If you don't liquidate, what stock trade will you need to do to become delta neutral again?
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Delta Hedging and Beyond
In-Class Problems
Answers
4. If you unwind at the new prices your profit is:
P&L on Calls 100,000 x (0.30 – 1.25)
Stock -29,000 x (95 - 100)
Total
= -95,000
= +145,000
= +50,000
5. If you wanted to rehedge, with the new delta, you should only be short
100,000 x 0.10 = 10,000 shares.
You have to buy back 19,000 of the shares you shorted.
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Delta Hedging and Beyond
Hedging Greek Letter Risks
Derivatives risk management begins with the basic delta-neutral hedge, but it does not end
there. Serious derivatives users try to minimize the (unintended) exposure of their positions
to all sources of price variability, and at the lowest possible cost. This includes:
•
changes in the price of the underlying asset (delta)
•
•
•
•
changes in the delta for large asset price changes (gamma)
changes in volatility (vega)
time decay (theta)
changes in interest rates (rho)
and quite probably other things as well.
Each of these Greek letters is a partial derivative with respect to some parameter of the
option pricing equation (a calculus derivative, that is).
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Delta Hedging and Beyond
Principles of Generalized Hedging
1. The Greek letter risk exposures "add up." The total risk exposure of a position is the sum
of the exposures of the component securities.
•
To hedge a given type of risk fully, the aggregate exposure of all hedging
instruments must be equal in magnitude and opposite in sign to the aggregate
exposure to that risk for the position being hedged.
2. In general you need at least one hedge instrument per type of risk.
•
For example, to hedge both delta and gamma, you need a minimum of two hedge
instruments.
3. Not every instrument can be used for every type of risk.
•
•
For example, the bond can’t hedge delta, and neither the bond nor the stock can
hedge gamma.
Two options may have the same value for two different Greek letters (for example,
a European and call and put with the same maturity and strike have the same
gamma and the same vega). In that case, you can’t use the two options to hedge
those two risks separately.
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Delta Hedging and Beyond
Principles of Generalized Hedging, p.2
4. If there are more hedge instruments than risks to be hedged, the solution is not unique.
This allows optimizing on other aspects of the hedge. Things to optimize on include
•
•
•
•
minimizing the overall cost
maximizing the (theoretical) expected profit from selling overvalued options and
buying undervalued ones
minimizing the amount of future rebalancing that will be required
etc.
Such goals can be pursued using linear programming techniques.
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Delta Hedging and Beyond
Designing and evaluating option positions requires use of a software
implementation of an option pricing model. Rather than use a canned package
like the one that comes with the Hull textbook, we have an Excel spreadsheet with
the Black-Scholes option pricing model laid out in a few cells. This allows you to
develop your own customized analysis tool.
[Tricks you can do with the Option Calculator Spreadsheet]
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Extending Black-Scholes and Option Replication
Adjusting the Black-Scholes Equation for Dividend Payout
The original Black-Scholes model applies to options on a non-dividend paying stock.
But many underlying assets make cash payouts. It is not difficult to adjust the B-S
equation. In addition to dividend-paying stocks, this also leads to option pricing
equations for many other underlying assets, including
•
•
•
•
•
•
stock indexes
foreign currencies
futures
interest rates
commodities
and lots of other things
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Extending Black-Scholes and Option Replication
Adjusting the Black-Scholes Equation for Dividend Payout
Discrete dividends: Suppose the stock will pay a dividend of $D and it goes ex-dividend on
date tDIV (which is sometime prior to option expiration day).
Replace the stock price S in the formula by S*
S* = S −
Call option value:
C = S * N[ d
]
De
−r t DIV
[
− X e − rT N d − σ T
]

σ2 

ln S * / X + r +
T


2


where d =
σ T
Call delta:
δ CALL = N [ d
]
Multiple discrete dividends: The adjustment is the same. S* is set equal to S minus the
present value of all dividends to be paid over the option's lifetime.
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Extending Black-Scholes and Option Replication
Early Exercise of American Calls
An American call should not be exercised early, in theory, except possibly just before the
underlying goes ex-dividend (early exercise gives away the option's time value).
• Even if you think the stock is about to go down, sell the option, don't exercise it.
(Unfortunately, this works in theory but not in practice! In the real world, the best
available bid in the market may be below intrinsic value. Exercise in that case.)
Theory shows that it may be rational to exercise an American call just before ex-dividend
day. Exercise if the intrinsic value is more than the value of a European call with the same
maturity and strike price, but at a stock price that is below the current price by the amount
the stock price will fall when it goes ex-dividend.
• Exercise if St - X > CEUR(St - div, X, T-t)
(assuming the stock falls by the full amount of the dividend)
Early exercise is more likely with
• high intrinsic value
big ex-dividend price drop
• large dividend
• short time remaining to maturity
• low volatility
low remaining time-value
• low interest rate
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Extending Black-Scholes and Option Replication
Adjusting the Black-Scholes Equation for Dividend Payout
Continuous proportional payout (the Merton Model): Suppose the underlying is like a
stock index portfolio that is most easily modeled as paying a dividend flow at a continuous
proportional rate q. For example, if the annual dividend yield on the S&P 500 Index is
currently 2.5%, set q = 0.025.
Replace the stock price S in the formula by S*:
S*
Call option value:
= Se
C = S e − q T N[ d
]
− qT
[
− X e − rT N d − σ T
]

σ2 

ln S / X + r − q +
T


2 

where d =
σ T
Call delta:
Note the change in delta!
FINC-UB.0043 Futures and Options Spring 2018
δ CALL = e − q T N [ d
]
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Extending Black-Scholes and Option Replication
Adjusting the Black-Scholes Equation to Price Options on Other Assets
European options on other underlying assets can be priced using the suitable variant of the
Black-Scholes equation. The formulas are simply the Merton continuous dividend equation
with appropriate values for q.
Options on Foreign Currencies (Garman-Kohlhagen Model): The price of the underlying
is the exchange rate (in $ per unit of FX). The underlying pays interest at the foreign riskless
rate, so set q = rFOR. The riskless rate r is the domestic rate.
S*
Replace the stock price S in the formula by S*:
Call option value:
C = S e − r FOR T N[ d
]
−r
T
= S e FOR
[
− X e − rT N d − σ T
]

σ2 

ln S / X + r − rFOR +
T


2


where d =
σ T
Call delta:
FINC-UB.0043 Futures and Options Spring 2018
δ CALL = e − r FOR T N [ d
]
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97
Extending Black-Scholes and Option Replication
Adjusting the Black-Scholes Equation to Price Options on Other Assets
Options on Futures (Black 1976 Model): The underlying is the futures contract, so S in
the equation is the futures price, call it F. Since a position is taken in the underlying without
any cash having to be invested, the value for q is the riskless interest rate: Set q = r .
Replace the stock price S in the formula by the discounted value of the futures price F:
F e −r T
Call option value:
C = F e − r T N[ d
where d =
Call delta:
FINC-UB.0043 Futures and Options Spring 2018
]
ln F / X
[
− X e − rT N d − σ T
+
]
σ2
T
2
σ T
δ CALL = e − r T N [ d
]
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98
Extending Black-Scholes and Option Replication
Adjusting the Black-Scholes Equation to Price Options on Other Assets
Options on Short Term Interest Rates, or other Noninvestible Assets ("Black Model"):
Like futures, cash is not invested in the underlying. The Black futures option model is
typically used. The riskless rate r here is the current short term rate.
Replace the stock price S in the formula by the discounted value of the interest rate R that is
the underlying for the option (e.g., 3 month LIBOR at option maturity one year from today):
R e −r T
Call option value:
C = R e − r T N[ d
where d =
Call delta:
FINC-UB.0043 Futures and Options Spring 2018
[
]
− X e − rT N d − σ T
ln R / X
+
]
σ2
T
2
σ T
δ CALL = e − r T N [ d
]
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99
Extending Black-Scholes and Option Replication
Adjusting the Black-Scholes Equation to Price Options on Other Assets
Commodity Options:
•
Options on commodity futures are priced like other futures options.
•
Options on physical commodities like wheat must take account of the carrying
costs for the underlying.
° Fixed costs (e.g., an inspection fee, the cost of transporting the deliverable
commodity to the delivery point, etc.) should be handled like negative discrete
dividends, that is, S* equals S plus the present value of the fixed costs.
° Continuous charges (e.g., daily storage costs) should be treated like negative
continuous dividends, that is,
S* = S e+qT
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100
Extending Black-Scholes and Option Replication
Adjusting the Put-Call Parity Equation for Payouts
The Put-Call parity equation is altered when the underlying asset makes cash payouts.
The price of the underlying, S, is replaced in the put-call parity formula by S* as
defined above for the particular underlying.
•
discrete dividends:
C - P = S - PV(divs) - PV(X)
•
continuous payout:
C - P = S e -q T - PV(X)
•
foreign currencies S = exchange rate:
C - P = S e -Rfor T - X e -Rdom T
•
futures:
C - P = F e -r T - X e -r T
= e -r T ( F - X )
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Extending Black-Scholes and Option Replication
Put Valuation with Payouts
The fair values for put options for each of the above cases come from put-call parity.
The put delta is also affected in many cases. (The variable d in these formulas is
defined as it is in the appropriate call pricing equation shown earlier.)
Discrete dividends: P = C - S + PV(divs) + PV(X)
Put delta: δput = N [ d ] - 1
Continuous payout: P = C - S e -q T + PV(X)
Put delta: δput = e -q T N [ d ] - e -q T
Foreign currencies (S = exchange rate):
Put value: P = C - S e -Rfor T + X e -Rdom T
Put delta: δput = e -Rfor T N [ d ] - e -Rfor T
Futures:
P
= C - F e -r T + X e -r T
Put delta: δput = e -r T N [ d ] - e -r T
= - e -r T N [ - d ]
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Extending Black-Scholes and Option Replication
Options on Futures
Many futures contracts have associated options trading on the same exchange.
Active markets
•
Agricultural: Corn, soybeans, wheat, cotton, sugar
•
Oil: Crude oil, natural gas
•
Gold
•
Interest rates: T-Bonds, T-Notes, Euro$, Bund
•
S&P 500 and other stock indexes
•
VIX volatility index
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Extending Black-Scholes and Option Replication
Unique Features of Futures Options
The underlying is a futures contract
•
•
exercise of a futures call leaves you long a futures contract, that is immediately
marked to market
exercise of a futures put leaves you with a short position in the future
Pricing model
•
the "Black 1976" variant of Black-Scholes, with continuous dividend set equal to
the riskless interest rate
Simpler delivery process
•
•
no issue of what is cheapest to deliver
futures options are more liquid and easier to hedge than options on many spot
commodities and assets
American exercise
•
American futures options (which most are) will be exercised early, both puts and
calls. However, the difference in value between American and European options
with the same terms is much smaller for futures options than for stock options.
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Extending Black-Scholes and Option Replication
Unique Features of OTC Foreign Currency Options
Price quotes are in terms of "vols," i.e., the volatility parameter to be input into the GarmanKohlhagen pricing equation.
"Moneyness," if expressed in dollar terms, is quoted relative to the forward rate.
• For example, if the spot rate on the Euro is 1.35 and the three month forward is
1.36, a three month "at the money forward" (ATMF) call will be struck at 1.36.
Usually, though, an option's moneyness is defined in terms of its delta.
• A "25-delta" call, will be one whose delta according to the formula is 0.25. A 25delta put has delta of -0.25.
Quotes (in vol terms) are often given on combination positions:
• Straddle: Long a call and a put with the same maturity, both struck at the money
forward.
• Strangle: Long a 25-delta call and a 25-delta put (i.e., both are out of the money).
• Risk Reversal: Long a 25-delta put, short a 25-delta call (also called a "Range
Forward" or a "Collar").
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Extending Black-Scholes and Option Replication
Structuring Portfolio Payoffs: Creating a Protective Put Position
Investors particularly like a payoff pattern that resembles a protective put position:
unlimited upside potential but a floor on the downside. They will pay a premium for
structured products that offer this pattern.
Portfolio
Value
Original
Portfolio
V0
Protected
Portfolio
I0
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Stock Market
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106
Extending Black-Scholes and Option Replication
Buying Put Protection with Options on a Specific Portfolio
Securities firms will create and sell options that their customers want to buy, including a put
option on a specific portfolio.
Consider buying a protective put on a portfolio that is currently worth V0 = $100 million.
The objective is to place a floor of $96 million on the total portfolio value in 1 year.
The parameters of the problem are:
V0 = 100
Floor = 96
Horizon = 1 year
Riskless interest = 8.00%
Portfolio volatility = 0.20
No dividends (dividends to be received are simply included in the
value of the portfolio to be guaranteed)
There are several different ways to set the problem up. It is important to distinguish clearly
between changes in the amount of stock in the portfolio from purchases and sales, versus
changes in the prices of the stocks. We only want to be protected against price changes.
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Extending Black-Scholes and Option Replication
Buying Put Protection with Options on a Specific Portfolio, p.2
The underlying asset for the put option is the whole portfolio, whose value V is assumed to
follow the standard Black-Scholes lognormal diffusion. That is, one put is based on one
"share" of portfolio, whose price is the portfolio's current market value. Setting the asset
value V0 = 100, and the put strike price equal to the desired floor, X = Floor = 96, the
Black-Scholes value for the put is
P(V0, X) = P(100, 96) = $3.230 million
Buying this put would guarantee that if the value of our $100 million portfolio is below $96
million a year from now, the protective put will make up the difference.
A problem with this solution is that it costs more than we have to invest:
Cost = V0 + P(V0, X) = 100 + 3.230 = 103.230. If we sell off some stock to buy the
put, we don't have $100 million invested anymore. A 96 strike put is not so far out of the
money relative to our new smaller portfolio, so it will cost more than what we just
calculated.
It is not hard to find the solution to this problem by iteration. We will find that if $100
million is all we can invest, the protective put that gives a floor of 96 will cost $4.506
million and we will only have $95.494 left to invest in stocks.
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Extending Black-Scholes and Option Replication
Creating the Protective Put Payoff Synthetically with a
"Portfolio Insurance" Strategy
A big problem with buying a put on a specific portfolio is that such options are not widely
traded. We have to find a securities firm willing to write the puts, and then negotiate an
acceptable price. (This used to be a bigger problem than it is today.)
In the late 1970s, two Berkeley finance professors, Hayne Leland and Mark Rubinstein,
proposed, and began marketing, an idea they called "Portfolio Insurance." It amounted to a
dynamic trading strategy that would replicate the desired protective put payoff, simply by trading
between the original portfolio and riskless bonds.
The concept is simple. The basic replicating strategy for any option position is simply to hold an
amount of the underlying that will have the same delta as the desired position and to borrow or
lend enough of the riskless asset so that the total replicating portfolio has the same cost. THIS IS
A KEY POINT TO REMEMBER!
Important: The following two steps give the general procedure to replicate an option:
1. Use the underlying (or some other derivative tied to the underlying, such as a
futures contract) to produce the same delta as the option one is trying to replicate.
2. Use riskless borrowing or lending to make the cost of the replicating position equal
to the theoretical price of the option.
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109
Extending Black-Scholes and Option Replication
Creating the Protective Put Payoff Synthetically using "Portfolio Insurance"
We have calculated that for our example, the amount of stock to keep, Vp, is
Vp = V0 - P(Vp,X) = 100 - 4.506 = 95.494 million
For the purpose of calculating a delta, we treat our stock position Vp as being one "share" of
portfolio with a current price of $95.494 million.
•
Using the Black-Scholes model, with S = 95.494, the put's delta is -0.3233.
•
Just as if it were 1 share of stock, the delta of Vp is 1.0.
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110
Extending Black-Scholes and Option Replication
Creating the Protective Put Payoff Synthetically using "Portfolio Insurance," p.2
The replicating portfolio for the put alone is
•
•
•
short 0.3233 units of Vp: -0.3233 x 95.494 = -$30.873 million of Vp
lend at the riskless rate to make the total out of pocket cost = the price of the put we
are trying to replicate, which is 4.506:
==> lend 4.506 + 30.873 = $35.379 million.
The overall portfolio insurance protected portfolio is therefore:
•
•
$95.494 - $30.873 = $64.621 million of Vp (sell off $35.379 million of the original
stocks)
$35.379 million in riskless bonds
This position will have to be rebalanced regularly to maintain the right delta as stock prices
change. This may be hard to do. It requires selling stocks to reduce effective market
exposure when stock prices are falling, and buying stocks when prices are rising. (Ask
about what happened on October 19, 1987.)
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Extending Black-Scholes and Option Replication
A Hybrid Security for Retail Investors: The Equity-Linked CD
An equity-linked CD is a type of certificate of deposit, whose payoff is tied to the stock
market, but with downside protection of principal.
Typical structure: The deposit has a maturity of five years, during which time it pays no
interest. At maturity, the initial deposit is returned, plus interest equal to the percentage
increase in the level of the S&P 500 stock index over the five years. If the S&P goes down,
the initial principal is returned in full but there is no additional interest payment.
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Extending Black-Scholes and Option Replication
Designing an Equity-Linked CD
Suppose you have the following data:
S&P index
S&P dividend yield
S&P volatility
Issuing bank's normal interest rate
on a 5 year zero coupon CD
1000
2.9%
.137
5.30%
1) Per $100 invested, what does it cost the bank in present value terms to provide this
payoff structure?
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Extending Black-Scholes and Option Replication
Designing an Equity-Linked CD
To get downside protection of principal, buy a five year zero coupon bond with 100 face
value. At 5.30 percent interest,
5-year zero, $100 face value = 100 / (1.0530)5 = 77.24
To get the upside price appreciation on the S&P, buy a 5-year call option.
• The underlying is $100 worth of the S&P portfolio. As before, think of this as "one
unit of an underlying asset whose current price is 100."
• The call should be at-the-money, so set the strike to 100.
(Note: Because we are doing everything in terms of rates of return, the current level of the
S&P is not used in solving this problem.)
5-year at the money call on $100 of the S&P 500 = 15.39
This leaves an immediate profit to the bank:
Profit = Deposit - bond price - call price = 100 – 77.24 – 15.39 = 7.36
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Extending Black-Scholes and Option Replication
Designing an Equity-Linked CD
$7.36 per $100 deposited is a high profit rate. Competition will surely drive it down.
Suppose the bank limits its fee to 2 percent of principal, taken out at the beginning. The
remainder of the deposit is then used to lock in the minimum payment at maturity and to
provide equity market exposure.
One alternative would be for the account to pay a small interest rate on the deposit, in
addition to 100 percent of the appreciation of the stock index.
Alternative #1: Full matching of the S&P on the upside plus payment of a fixed interest rate
if the S&P falls.
Deposit
100
less cost of call option
-15.39
less bank profit
- 2
equals amount available
to buy zeros
= 82.61
Buying $82.61 of 5-year zeros produces 82.61 x 1.05305 = $106.94 at maturity. This
corresponds to:
annual interest rate
= 1.35%
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Extending Black-Scholes and Option Replication
Designing an Equity-Linked CD
Alternatively, the bank can limit its fee to 2 percent of principal and use the extra funds to
give greater than 100 percent of the appreciation on the index, if it goes up.
Alternative #2: Guaranteed return of principal without interest if the market goes down, and
more than 100% of the S&P capital gain if it goes up.
Deposit
less cost of 5-year zeros
less bank profit
equals amount available
to buy calls
100
-77.24
- 2
= 20.76
By buying calls on more than $100 worth of the S&P 500, the bank can guarantee return of
$100 principal plus
20.76 / 15.39 = 134% of the S&P capital gain.
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116
Derivatives Advanced Topics: Interest Rate Derivatives and Interest Rate Models
Some of the most important and most actively used derivatives are those based on interest
rates. These allow management of interest rate risk on loans and bonds.
We first review how interest rate futures and forwards work and how to set up hedges with
these contracts. We then go on to consider interest rate swaps and to extend the ideas from
option valuation to interest rate derivative products with option features such as caps and
floors.
The most basic model for interest rate options is the Black model, a variant of BlackScholes. Unfortunately, the empirical evidence shows that real world interest rate processes
are considerably more complicated than simple logarithmic diffusions.
We discuss the problems with the Black model framework and look briefly at alternative
models of the interest rate process. As with other instruments, interest rate models come in
both equilibrium and arbitrage-free formulations, with pros and cons attached to each.
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Interest Rate Derivatives and Interest Rate Models
Important Concepts in this Section
•
•
•
•
•
•
•
•
Types of interest rate derivatives
LIBOR and the Eurodollar futures contract
Hedging with FRAs and Eurodollar futures
Swaps
The Black model for interest rate options
How interest rate caps, floors, and collars work
Problems with the Black model for pricing interest rate derivatives in the real world
Alternative pricing models for interest-dependent securities
° equilibrium models (e.g., Vasicek model)
° arbitrage-free models (e.g., the LIBOR Market Model)
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Interest Rate Derivatives and Interest Rate Models
Main Types of Interest Rate Derivatives
The underlying is always an interest rate applied to a "notional" principal amount for a
specified time period ("tenor").
The simplest interest rate derivatives are basic forward and option contracts, with a single
maturity date.
Forward Rate Agreement (FRA): A forward rate agreement is a kind of forward contract.
A FRA fixes the interest rate to be paid on the notional principal at a specified strike value.
The payment period (the tenor) begins on the contract's maturity date. If the market rate on
that date is above the strike rate, the long FRA counterparty receives a payment from the
short equal to the difference in interest cost between the two rates. If the market rate is
lower than the strike rate, the long pays the short the difference in interest.
Interest rate call (or put) option, "caplet" (or "floorlet"): Like a FRA, because it has a
single maturity date, but the payoff is like an option: If the market rate is above the strike at
maturity, the call buyer receives the difference from the writer, but if the market rate is
below the strike, the option expires worthless. A "caplet" is a single call option in a cap
contract, and a "floorlet" is a single put in a floor.
Examples: If the strike interest rate is 5% on a 3 month FRA or call option with $1 million
notional principal and 6 month tenor, and the actual 6-month rate 3 months from today is:
6%: The FRA and the call both receive (.06-.05)(1/2)(1,000,000) = $5000
4%: The FRA pays (.04-.05)(1/2)(1,000,000) = -$5000; there is no payoff on the call.
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Interest Rate Derivatives and Interest Rate Models
Main Types of Interest Rate Derivatives
The most important interest rate derivatives involve repeated payments at regular intervals
over time. They are like a set of FRAs or options with the same terms and sequential
maturities.
Swap: A swap consists of a series of FRAs with the same strike rate and periodic maturities
(e.g., every 3 months) . A swap is useful for turning a loan with a fixed interest rate into one
with a floating rate tied to the underlying rate for the swap, or vice versa.
Cap, Floor: Like a swap, a cap (floor) contract is a series of interest rate calls (puts) with
the same strike and sequential maturities. A cap can be used to place a maximum on the
interest rate one has to pay on a floating rate loan, without locking in that rate if the actual
market rate turns out to be lower. A floor can be used by a floating rate lender to lock in a
minimum rate that will be received.
Swaption: An option to enter into a swap at a swap rate equal to the strike of the swaption.
A "2 by 5" swaption, is a two-year option to enter into a 5-year swap.
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Interest Rate Derivatives and Interest Rate Models
The next several slides review hedging a single future cash flow with interest rate futures or
FRAs.
The most basic interest rate derivative is the forward rate agreement (FRA). A FRA fixes the
level of some interest rate, such as 90-day LIBOR, to be paid on the notional principal at a
specified strike value.
The Eurodollar futures contract is effectively the same thing, except that it is marked to
market daily. We will see that setting up a hedge correctly with FRAs can be easy but
hedging with Eurodollar futures becomes a little trickier than one might first imagine.
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Interest Rate Derivatives and Interest Rate Models
Recall that:
•
Like other short term "money market" rates, LIBOR is quoted on a 360 day year. If the
quoted rate is 2.00%, interest accrues at the rate of 2.00%/360 per calendar day. Interest is
not compounded when the holding period is a year or less.
•
At 2.00%, a loan of 100 for 90 days would earn interest of (90/360) x (.02) x 100 = $0.50.
A one year loan would pay (365/360) x (.02) x 100 = $2.028.
•
The Eurodollar futures price is defined by: F = (100 – Annualized Forward LIBOR Rate),
the underlying is 90-day LIBOR, and the notional is $1 million.
•
This makes the "dollar value of a basis point" (called DV01) equal $25 per contract. If the
Eurodollar futures price goes from 98.20 to 98.25, this corresponds to the annualized
forward interest rate falling 5 basis points, from 100 - 98.20 = 1.80% to 1.75%. The long
position would get a 5 x $25 = $125 mark-to-market cash inflow. The short would lose
$125.
•
Eurodollars are NOT Euros. They are deposits in non-U.S. banks that are denominated in
dollars. Originally, Eurodollar deposits were at banks in London; now they can be
anywhere.
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Interest Rate Derivatives and Interest Rate Models
Eurodollar Futures September 2, 2016
Chicago Mercantile Exchange
Underlying instrument
• Special index of 90 day Euro$ deposit rates
(LIBOR)
Month
Futures Prices
• Quoted as 100 minus interest rate
• Tick = 0.01 = $25.00 (half ticks are used now because
rates are so low, and quarter ticks for near maturities.)
Quantity
• $1 million ("notional principal")
Expiration dates
• Monthly for next 4 months, then every March, June,
September, December
• 2 London business days before 3rd Wednesday of the
expiration month.
• Contracts currently traded for maturities up to 10
years.
Delivery
• Cash settlement only
• No delivery options
FUNC-UB.0043 Futures and Options Spring 2017
Open
High
Low
Last
Change
Last Updated: Friday, 02 Sep 2016 02:30 PM
16-Sep
99.1275
99.17 99.1225
99.14
16-Oct
99.1
99.135
99.095
99.105
16-Nov
99.09
99.11
99.085
99.09
99.115
16-Dec
99.06
99.05
99.055
17-Jan 99.0900B 99.0500A
17-Feb 99.0750B 99.0300A 99.0300A
17-Mar
99.02
99.085
99
99.01
17-Jun
98.975
99.055
98.955
98.965
99.015
17-Sep
98.93
98.915
98.925
17-Dec
98.885
98.97
98.865
98.885
18-Mar
98.865
98.945
98.84
98.86
18-Jun
98.835
98.915
98.805
98.83
98.795
98.885
98.77
98.795
18-Sep
18-Dec
98.75
98.84
98.725
98.755
19-Mar
98.725
98.8
98.7
98.73
19-Jun
98.69
98.765
98.665
98.695
19-Sep
98.655
98.735
98.625
98.66
19-Dec
98.615
98.695
98.58
98.615
20-Mar
98.59
98.64
98.55
98.585
20-Jun
98.555
98.615
98.51
98.55
20-Sep
98.52
98.575
98.475
98.51
20-Dec
98.48
98.54
98.43
98.47
21-Mar
98.445
98.49
98.395
98.43
21-Jun
98.41
98.465
98.355
98.395
21-Sep
98.355
98.41
98.32 98.3550A
21-Dec
98.31
98.36
98.28 98.3150A
22-Mar
98.28
98.355
98.245
98.285
22-Jun
98.27
98.315
98.215
98.25
22-Sep
98.26 98.2700B
98.185
98.225
22-Dec
98.21
98.23
98.155
98.185
23-Mar
98.21
98.215
98.135 98.1700B
23-Jun
98.165 98.1850B
98.11
98.14
23-Sep
98.12
98.175 98.1100A 98.1100A
Part I: Forwards and Futures
0.01
0.005
UNCH
-0.01
-0.005
-0.005
-0.01
-0.005
UNCH
UNCH
0.005
0.005
0.005
0.01
0.01
0.01
0.01
0.005
0.005
0.005
0.005
UNCH
-0.005
-0.01
-0.01
-0.01
-0.01
-0.015
-0.015
-0.015
-0.015
-0.02
-0.02
Settle
99.1375
99.1
99.09
99.055
99.04
99.03
99.01
98.97
98.93
98.885
98.865
98.835
98.8
98.76
98.735
98.705
98.67
98.625
98.595
98.56
98.525
98.48
98.445
98.405
98.365
98.325
98.295
98.26
98.23
98.2
98.18
98.15
98.125
©2017 Figlewski
Estimate
d Volume
Prior Day
Open
Interest
368,670
29,020
12,240
401,421
0
0
289,682
219,701
214,153
263,645
142,638
140,166
135,613
129,927
90,655
88,272
57,454
62,491
32,079
38,811
24,590
23,916
17,797
21,984
2,132
1,717
1,281
1,264
134
135
130
85
42
1,084,257
134,713
26,805
1,522,347
150
0
1,102,568
1,011,804
860,533
1,332,003
635,236
493,443
456,728
616,586
410,653
319,744
246,531
266,208
144,450
101,479
80,262
101,896
55,127
54,261
24,659
18,623
11,884
6,558
4,916
5,616
4,937
840
1,365
123
Interest Rate Derivatives and Interest Rate Models
Example: Hedging the Repricing of a Swap Payment
with Eurodollar Futures
Suppose your firm is paying a fixed rate of 2.70 percent and receiving 6 month US dollar
LIBOR on a $50 million swap. Repricing is every 6 months.
[What is an interest rate swap? A swap is a contract in which periodically (e.g., every 6
months), the two counterparties exchange ("swap") two cash amounts calculated as the
interest for that period on a given "notional" principal (e.g., $50 million) using two different
interest rates. Generally one rate is fixed and the other is floating (e.g, 2.70% fixed annual
rate versus 6 month LIBOR).]
At next March's repricing, the floating rate will be reset to the level of 6 month LIBOR in
the market on that date. You want to use Eurodollar futures to hedge the interest rate risk on
the swap payment that will be based on that rate.
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Interest Rate Derivatives and Interest Rate Models
Example: Hedging with Eurodollar Futures, p.2
An important first question
Considering the risk and the instruments involved and how the Eurodollar futures contract
works, do you want to buy Eurodollar futures or sell Eurodollar futures?
Is figuring out the answer to this obvious question harder than you might have thought? One
way to unravel the complexity is to apply "Figlewski's Rule."
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Interest Rate Derivatives and Interest Rate Models
"Figlewski's Rule"
A Rule of Thumb for Avoiding Really Stupid Mistakes in Hedge Design
To avoid selling futures when you really ought to buy them, or buying futures when you
really should sell, break the thought process into two parts:
1. Figure out what you are afraid might happen that will hurt the position you want to
hedge.
Then,
2. Take a futures position that will make money if what you are afraid of in step 1
actually does happen.
How does Figlewski's Rule work in this case?
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126
Interest Rate Derivatives and Interest Rate Models
Example: Hedging with Eurodollar Futures, p.3
How many contracts should you trade?
A Eurodollar futures contract = $1 million principal value and the swap notional principal is
$50 million. Do you trade 50 contracts?
No. We need "dollar equivalence". Since the repricing interval is 6 months, a 1 basis point
change in 6 month LIBOR translates to a dollar change in the floating payment equal to
DV01 = .0001 x (180 / 360) x $50,000,000 = $2500
A 1 basis point change in the Eurodollar futures price is
.0001 x (90 / 360) x $1,000,000 = $25 per contract
To achieve dollar equivalence, so that the futures hedge offsets the change in dollar value of
the swap payment, you need to trade
$2500 / 25 = 100 contracts
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Interest Rate Derivatives and Interest Rate Models
In setting up a simple interest rate hedge, there are three relevant dates:
•
today,
•
the date on which the cash flow you are trying to hedge will occur,
•
and the date on which the uncertainty over that cash flow is resolved.
Dollar equivalence requires that the cash flow on the hedge position should be equal in size
and opposite sign, as of the same date. Getting this right when the cash flow and the
resolution of uncertainty are on different dates involves present-valuing or future-valuing the
cash flow from the hedge to get it to match up at the same time with the cash flow being
hedged.
Futures and forwards are basically the same kind of contract, but because futures are marked
to market every day, their cash flows begin immediately as soon as the interest rate changes,
while a forward contract does not pay until it reaches maturity. (There might be adjustments
in the collateral requirements for the FRA, but this doesn't involve cash payments to the
counterparty.)
This key difference leads to different hedge design for the two.
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Interest Rate Derivatives and Interest Rate Models
Here are futures quotes for the next 8 quarters and the forward interest rates extracted from
those futures quotes . The discount function computed from these rates, PV($1), is used for
discounting future cash flows.
Spot interest rate: r0 = 5.00%
Notional: V = $100,000,000
t0
t1
t2
t3
t4
t5
t6
t7
t8
years to maturity
interval Dt
0
0.25
0.25
0.25
0.5
0.25
0.75
0.25
1
0.25
1.25
0.25
1.5
0.25
1.75
0.25
2
Futures price
rt
PV($1)
95.00
5.00%
1
94.75
5.25%
0.98765
94.50
5.50%
0.97486
94.25
5.75%
0.96164
94.00
6.00%
0.94801
93.75
6.25%
0.93400
93.50
6.50%
0.91963
93.25
6.75%
0.90492
93.00
7.00%
0.88991
rt plus 1 b.p.
5.01%
5.26%
5.51%
5.76%
6.01%
6.26%
6.51%
6.76%
7.01%
PV($1 at rt + 1b.p.)
1
0.98763
0.97481
0.96157
0.94792
0.93388
0.91949
0.90477
0.88973
To compute DV01s for a 1 basis point change in the interest rate, we consider two
possibilities: either the rate changes for just one future period and all the others stay the
same, or else the whole yield curve moves and all future rates go up a basis point.
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Interest Rate Derivatives and Interest Rate Models
Hedging with a FRA
Hedging the quarterly interest payment on a floating rate loan that will occur on date t4.
At t4 the cash flow will be:
At the current forward rate this is:
(notional) x (rate at t3) x (interval from t3 to t4)
100,000,000 x 5.75% x 0.25 = $1,437,500
WHEN IS THE UNCERTAINTY RESOLVED? At t3 when the interest rate that determines
the size of the interest payment is set. So we need our hedge to mature at t3.
Suppose the rate at t3 goes up 1 b.p.: 100,000,000 x 5.76% x 0.25 = $1,440,000
The DV01 as of t4 is therefore: $1,440,000 - $1,437,500 = $2500.
To offset the risk, hedge with a $100 million FRA that fixes a rate for the period from t3 to
t4. But if the FRA's cash flow occurs at t3, the timing of the cash flows doesn't match up.
Real world FRAs are often designed so that a perfect hedge of the interest payment is
possible. When date t3 arrives, the payoff on the FRA is set equal to the present value of
(rt3 – s), where s is the strike rate on the FRA. The discounting is done at the t3 market rate
rt3. That way the cash flow on the FRA exactly offsets the extra interest above the strike rate
s that is caused by the realized rate rt3
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Interest Rate Derivatives and Interest Rate Models
Hedging with Eurodollar Futures
Hedging the quarterly interest payment on a floating rate loan that will occur on date t4.
At t4 the cash flow will be:
100,000,000 x 5.75% x 0.25 =
$1,437,500
The uncertainty is resolved at t3 so we use the futures contract that matures at t3 (or
immediately after).
The DV01 on the loan payment as of t4 is :
$2500.
The DV01 on a Eurodollar futures contract (as of t0 ) is : $25.
If the futures price changes, the futures cash flow begins immediately. To bring the $2500
loan DV01 back to the present, multiply by the t4 discount factor 0.94801 to get
The DV01 on the loan payment as of t0 is :
The DV01 on a Euro$ future is (as of t0 ):
$2500 x 0.98401 = $2370.
$25
Hedge the interest on the $100 million loan with:
(2370 / 25) = 94.8 ==> 95 t3 Eurodollar futures contracts.
The extra discounting needed when hedging with futures is called "tailing the hedge".
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Interest Rate Derivatives and Interest Rate Models
What is a Swap?
A Swap is an agreement between two counterparties to exchange periodic cash payments in
the future, based on some prespecified formula.
Key features:
• agreement between counterparties: a swap is a kind of over-the-counter derivative;
• cash flows are exchanged: both counterparties have a liability to pay (although only
the net difference actually changes hands);
• periodic: a swap normally entails a sequence of future payments.
• under Dodd-Frank regulations, swaps with standard features may be set up OTC,
but they now must be cleared through a Central Clearing CounterParty (CCP)
The most common type of swap is a fixed-for-floating interest rate swap.
(Note that in recent years, the word "swap" has come to be used more broadly than this. In
some contexts, a "swap" is just another term for a forward contract. Interest rate swaps are
as described in the next few slides.)
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Interest Rate Derivatives and Interest Rate Models
Example of a Swap
The counterparties A and B agree that every 6 months for the next 3 years, A will pay to B
the interest on a "notional" principal amount of $100 million at the fixed rate of 10%. B will
simultaneously pay to A the interest on the same notional $100 million at a floating interest
rate equal to 6-month LIBOR (as of the beginning of each 6 month period) plus 50 basis
points. In practice, the two cash flows are netted and the counterparty with the larger
liability simply pays the net difference to the other counterparty.
Important point
The $100 million notional principal never changes hands and is never at risk. Its purpose is
only to turn an interest rate into a dollar payment amount.
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Interest Rate Derivatives and Interest Rate Models
Sample Swap Payment Schedule
Date
A owes B
LIBOR
(%)
B's interest
rate
B owes A
Initial
—
8.00
—
t=6
months
$5 million
8.00
12 months
$5 million
18 months
Net Payments
A pays B
B pays A
—
—
—
8.50
$4.25
million
$0.75
million
—
8.50
8.50
$4.25
million
$0.75
million
—
$5 million
9.00
9.00
$4.50
million
$0.50
million
—
24 months
$5 million
9.50
9.50
$4.75
million
$0.25
million
—
30 months
$5 million
9.75
10.00
$5.00
million
—
—
36 months
$5 million
—
10.25
$5.125
million
—
$0.125
million
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Interest Rate Derivatives and Interest Rate Models
Why Swap?
A swap often seems to offer both counterparties lower borrowing costs than are available to
them otherwise.
Example: Suppose the following are the normal borrowing costs for A and B.
Floating rate
A - LIBOR + 120 b.p.
B - LIBOR + 100 b.p.
3 year fixed rate
A - 11.0%
B - 10.0%.
Firm A would like to borrow for three years at a fixed interest rate. The market would
charge a firm with A's credit quality 11% to do this.
Firm B would like to borrow for three years at a floating rate. The market rate for B would
be LIBOR + 100 basis points.
But if A borrows in the market at the floating rate and B borrows at the fixed rate, and they
then enter into a swap with each other in which A pays 10.0% fixed rate to B while B pays
LIBOR + 50 b.p. to A, they both can reduce their overall borrowing costs.
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Interest Rate Derivatives and Interest Rate Models
EFFECTIVE BORROWING COSTS BEFORE THE SWAP
A borrows fixed and B borrows floating
Floating Rate Market
Fixed Rate Market
A
A pays11% fixed rate
B
B pays LIBOR + 100 b.p.
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Interest Rate Derivatives and Interest Rate Models
How does this work?
A takes out a 3 year $100 million floating rate loan at LIBOR + 120 b.p. and enters into a
pay-fixed-receive-floating interest rate swap with B.
• The swap payments received from B will cover LIBOR plus 50 b.p.. A adds an
extra 70 b.p. and pays LIBOR plus 120 b.p. to its lender, and 10.0 percent to B.
• This effectively turns the floating rate loan into a fixed rate loan, with a total
interest cost equal to 10.70 percent.
B takes out a fixed rate 3 year $100 million loan at 10.0 percent and enters into a payfloating-receive-fixed interest rate swap with A.
• The 10.0% fixed swap payments from A cover B's interest payments to the market.
• The swap turns the fixed rate loan into a floating rate loan at LIBOR + 50 b.p. (paid
to A).
A saves 30 b.p. and B saves 50 b.p. in borrowing costs on 3 year financing.
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Interest Rate Derivatives and Interest Rate Models
EFFECTIVE BORROWING COSTS AFTER THE SWAP
A borrows floating, B borrows fixed, and they swap
Floating Rate Market
Fixed Rate Market
A pays LIBOR + 120 b.p.
(LIBOR + 50 b.p. comes
from B; A adds 70 b.p.)
A
In the swap, B pays
LIBOR + 50 b.p. to A
In the swap, A pays10%
fixed rate to B
B
B pays10% fixed rate
received from A
The result of swapping is that, effectively, A pays 10.70 percent fixed rate and B pays
LIBOR + 50 b.p.
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Interest Rate Derivatives and Interest Rate Models
Are Swaps a Free Lunch?
How can a swap offer a profit to both counterparties? The most logical explanation focuses
on the fact that, unlike a normal loan, the notional principal is never at risk.
An ordinary loan carries a default premium because the borrower may default and not repay
the principal. The premium for a long term loan is greater than for a short term loan, and the
difference is bigger the less creditworthy is the borrower.
The more creditworthy counterparty therefore has a comparative advantage borrowing at a
long maturity and the less creditworthy counterparty has a comparative advantage (that is, a
smaller disadvantage) borrowing at a short maturity.
In a swap, neither counterparty pays a premium for the risk that they will default on the
principal amount, so swapping allows them to exploit each one's comparative advantage and
divide the net improvement in borrowing terms between them.
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Interest Rate Derivatives and Interest Rate Models
Pricing a Swap
The Swap Rate: Like a forward price, the "swap rate" in the market is the fixed interest rate
for a given maturity at which a swap against LIBOR can be set up such that neither
counterparty has to make a payment to the other at the beginning.
How is the swap rate determined? The future cash flows on the swap can be thought of in
two equivalent ways:
1. A swap is a series of forward contracts (forward rate agreements, actually).
• The "strike price" is the fixed interest rate and the "underlying" is the floating rate. In our
example, the swap between A and B is like 6 forwards with sequential maturities; the strike on
each one is the $5 million fixed rate payment and the underlying is the floating market value of
1/2 year's interest on $100 million computed as market LIBOR 6 months before the payment
date, plus 50 b.p.
2. A swap is like being long one bond and short another with the same face value.
• Paying fixed and receiving floating gives the same cash flows as being long a floating rate bond
and being short (i.e., issuing) a fixed rate bond. In our example, A's position is the same as
issuing a $100 million face value 10 percent coupon 3 year bond and using the proceeds to buy
$100 million of bonds paying a floating interest rate of LIBOR plus 50 b.p. B's position is the
reverse, long the fixed rate bond and short the floating rate bond.
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Interest Rate Derivatives and Interest Rate Models
Pricing a Swap
Compare a swap to pay a fixed rate C and receive LIBOR on future dates t1, t2, t3 with the
payments on a series of FRAs all struck at rate C, and with payments on the bond portfolio
that is long a date t3 floating rate bond paying LIBOR and short a fixed rate bond with
coupon rate C.
Fixed for
Floating Swap
C
C
C
Pay
Receive
LIBOR(t0)
LIBOR(t1)
LIBOR(t2)
C
t1
FRA
t2
FRA
t3
FRA
Pay
Receive
Pay
Receive
LIBOR(t0)
C
LIBOR(t1)
Pay
Receive
C
LIBOR(t2)
100
Long floating
rate bond
Pay
Receive
Short fixed
rate bond
Pay
Receive
LIBOR(t0)
C
LIBOR(t1)
C
LIBOR(t2) + 100
C + 100
100
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Interest Rate Derivatives and Interest Rate Models
Pricing a Swap, p.2
The Value of a Swap
Like both of these positions, the value of a swap struck at the market swap rate is zero
initially. It will become positive or negative as market interest rates move. A rise in the
floating rate in the market increases the value of the swap for the fixed rate payer
(counterparty A), and lowers it for the receiver of the fixed rate (counterparty B). The
change in value is easy to compute from the effect on the two bond values.
Risk Exposures
The exposure to interest rate risk, as measured by duration, convexity, DV01, and other
sensitivities can be obtained easily by considering the equivalent bond or position in
forwards.
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Interest Rate Derivatives and Interest Rate Models
Variations on the Theme
The concept of swapping is very powerful and many new types of swaps and related
contracts have become commonplace
Currency swaps (e.g., dollars vs. Euro)
Asset swap (exchange payments on some asset against a floating riskless rate)
Amortizing swaps (notional principal varies over time, like a mortgage)
Yield spread swaps
- basis swaps (e.g., T-bill rate vs. LIBOR)
- yield curve swaps (e.g., 10 year rate vs. 3 month)
- diff swaps (US $ interest vs. Euro-zone interest)
Equity swaps (e.g., S&P return vs. fixed rate)
Commodity swaps (e.g., oil prices vs fixed rate)
Total return swap (can transform any instrument into a different one)
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Interest Rate Options and Interest Rate Models
Adjusting the Black-Scholes Equation to Price Interest Rate Options
The underlying is an interest rate so, like a futures contract, one does not invest cash to "buy
and hold" a position in it. The basic valuation model is a modified version of the
"Black '76" futures option model, so it is known as the Black model.
One important issue is that there is no longer a riskless interest rate. Still, discounting a cash
flow from option maturity at the (stochastic) interest rate is easily accomplished: simply
multiply it by the price of a zero coupon bond maturing on that date. This makes use of
today's price of a traded security to capture all of the uncertainty about the future course of
the discount rate. Because the zero coupon bond price can simply be observed in the
market, the stochastic behavior of the discount rate does not have to be modeled at all.
This is an example of an extremely useful and powerful technique for derivatives valuation,
known as a "change of numeraire." Here, the interest rate option's payoff is in dollars at
option expiration, but we change what we are thinking of as the unit of account--the
numeraire--from "dollars on date T in the future " (which we don't know how to present
value) to "date T maturity zero coupon bonds" (which the market is present valuing for us).
When it is possible, a suitable change of numeraire reduces the number of random factors
that have to be dealt with and can significantly simplify option pricing problems.
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Interest Rate Options and Interest Rate Models
The Black Model for Interest Rate Options
The underlying rate, call it R, is the value of a specified interest rate as of option maturity
date T, for example, 3-month LIBOR. The assumption is that today's forward rate for date
T, call it F, is the expected value as of today (time 0) of R on date T, that is, F = E0[RT].
The probability distribution for R on date T is assumed to be lognormal with expected value
F and standard deviation σ T
Discounting in the formula at a stochastic "riskless" rate is handled by replacing e-rT by
B(0,T), the time 0 market price for a zero coupon bond maturing at date T.
Call option value:
(
C = B(0, T) F N[ d
where d =
Call delta:
FINC-UB.0043 Futures and Options Spring 2018
ln F / X
]
[
− X N d−σ T
+
])
σ2
T
2
σ T
δCALL = B(0, T) N [ d
]
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Interest Rate Options and Interest Rate Models
The Black Model for Interest Rate Puts
The formulas for interest rate put options come directly from put-call parity (which is also
modified when the underlying is an interest rate).
Put-Call Parity:
Put option value:
C − P = B ( 0 ,T ) ( F − X )
(
P = B(0, T) − F N [ − d
where d =
Put delta:
ln F / X
]
+
[
+ X N − d+σ T
])
σ2
T
2
σ T
δPUT = − B(0, T) N [ − d
]
(These equations make use of the property that the normal distribution is symmetric, so
1 - N [d] = N [ -d ].)
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Interest Rate Options and Interest Rate Models
Problems with the Black Model
•
The "riskless" interest rate changes randomly and it is correlated with
changes in the price of the underlying.
•
Mean reversion in interest rates; they are not a random walk
•
The theoretical term structure of interest produced by the model is not
consistent with the current yield curve observed in the market
•
Because there is only one source of risk, some commonly observed
interest rate behavior is inconsistent with the model
° twists in the yield curve (e.g., short rates rise and long rates fall at the same
time)
° changes in curvature of the term structure
•
Forward rate volatility is different for different maturities
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Interest Rate Options and Interest Rate Models
Valuation Models for Interest-Dependent Securities
A large number of theoretical pricing models have been developed for bonds and
other interest-dependent securities, and derivatives based on them. These are
some of the most complex valuation models around.
As is the case for other kinds of derivatives, there are two basic types of models:
"equilibrium" models and "arbitrage-free" models. Both try to model the
evolution of interest rates, from which pricing equations for particular securities
can be derived.
In other words, all interest-dependent securities are treated as derivatives based on
the underlying interest rates.
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Interest Rate Options and Interest Rate Models
Valuation Models for Interest-Dependent Securities
Equilibrium interest rate models
•
the short term interest rate is assumed to follow a stochastic process with plausible
features (as in the Black-Scholes and Black '76 models)
•
long term rates (the term structure) are derived as functions of the short rate process
•
the short rate may depend on several random factors; typically, it is assumed to
revert towards a long term mean and have stochastic volatility, both of which might
follow their own stochastic processes
•
but, model values for actual bonds need not match their current prices in the market
•
An example: the Vasicek model: dr = K (µ - r) dt + σ dz
change
in rate
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adjustment
long term
mean
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Interest Rate Options and Interest Rate Models
Arbitrage-free Valuation Models for Interest-Dependent Securities
•
Traders do not like pricing models that say current market prices are wrong (i.e.,
that there are arbitrage opportunities available among existing bonds)
•
Arbitrage-free models take the current term structure of interest rates in the market
as an input and derive interest rate processes that are consistent with it.
•
The Ho-Lee model was the first arbitrage-free model of the term structure. It was
essentially a binomial model which assumed that over the next time step, the entire
yield curve could move to just one of two possible new shapes. It was a theoretical
breakthrough, but it was obviously limited in how the term structure could behave,
and had the unfortunate feature that interest rates could go negative in the model.
•
The Heath-Jarrow-Morton (HJM) family of models eliminated the problems of
the Ho-Lee formulation. These model the evolution of the whole market yield
curve of forward interest rates in such a way that there are no arbitrage
opportunities either in the current structure of interest rates or in future rates that
are possible within the model.
•
HJM is mathematically elegant, but quite hard to use in practice.
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Interest Rate Options and Interest Rate Models
Arbitrage-free Valuation Models for Interest-Dependent Securities
Interest rate products may be valued and risk-managed in practice using an equilibrium-type
model that is adapted to incorporate the current term structure.
One model that does this is Hull and White's "Extended Vasicek" model:
dr = K ( θ(t) - r) dt + σ dz ,
where θ(t) specifies an expected drift function for the short interest rate that can be
tweaked to produce expected values for future short rates that are consistent with
the current term structure observed in the market.
θ(t) is calibrated to current bond prices.
Unfortunately, volatility is a fixed parameter, so the model can match market prices for
bonds, but not for interest rate options, and also the rate can go negative.
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Interest Rate Options and Interest Rate Models
Arbitrage-free Valuation Models for Interest-Dependent Securities
A way to extend the Extended Vasicek model, that is common in practice, is the
Black-Karasinski model.
d log(r) = ( θ(t) - a(t) log(r) ) dt + σ(t) dz ,
One key difference from the Extended Vasicek model is that this model is written in terms
of the log of the interest rate, so the rate can't go negative. Another is that volatility and the
speed of mean reversion are now allowed to vary over time. This makes it possible to
calibrate the model to both the current yield curve and also the current volatility surface in
the market, even though there is still only one source of risk in the model, dz.
The fact that short term interest rates in a number of countries are now (Spring 2017)
negative causes lots of problems for many interest rate models, like this one.
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Interest Rate Options and Interest Rate Models
Arbitrage-free Valuation Models for Interest-Dependent Securities
LIBOR Market Model: The Brace, Gatarek, and Musiela (BGM) model has nearly
become the industry standard at this point. The model focuses on the short interest rate at
each one of a set of relevant future dates, in particular the future repricing dates {t1, t2, ...,
tM} for a given swap.
dfm = σm(t) fm dZm
where
fm is the forward rate as of date t for the future period beginning at date tm
σm(t) is the volatility of fm as of date t
dZm is the m-th element of an M-dimensional vector of Brownian motions, with
mean vector 0 and covariance matrix Ω
The key assumption is that each rate follows its own lognormal diffusion process--M
sources of risk for a swap with M payment dates--but they must satisfy several constraints:
• The current rates are consistent with the observed forward rates in the market
• The expected drift of each forward rate is zero. The forward rate is the expected value of the
future spot rate—the Expectations Model holds. This imposes a consistency condition between
the volatilities and the drifts for the LIBORs in the model.
Even so, there can be a lot of parameters to calibrate and stochastic variables to simulate in a
Monte Carlo analysis. Variants of the BGM model impose further constraints to reduce the
computational burden.
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Bonds and Mortgages
Bond Options
Most interest rate derivatives like swaps, caps, and floors are based on rates. But there are
also option contracts written on bonds and bond futures, as well as a variety of optional
features, like callability or convertibility that may be embedded in the bonds themselves.
Bond Option Contracts
•
•
exchange-traded and over the counter contracts
•
options on T-Bond futures are traded at the Chicago Board of Trade.
options on specific Treasury bonds are traded over-the-counter by government
bond dealers
Embedded Options
•
•
callable bonds
•
convertible bonds (some corporate bonds can be exchanged for shares in the issuing
firm at the bondholder's option)
mortgage prepayment option (the borrower's right to repay a mortgage early is a
kind of call option)
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Callability in Bonds and Mortgages
Callable Bonds
Yields on callable bonds are evaluated in terms of the "Option-Adjusted Spread"
To compute the Option-Adjusted Spread (OAS), first value all embedded options and subtract
the total value of the optionality from the bond's market price. Compute the yield to maturity
on this option-free price. The spread relative to the yield to maturity for the comparable
maturity Treasury bond is the OAS.
Example: PDQ Corporation has an outstanding bond issue with the following terms:
Maturity
11 years
Coupon
7.30 percent
Face value
100
Callable at a price of 103, beginning in year 5
Current market price
92.00
Quoted yield (at P = 92.00)
8.43 percent
Yield on 11 year Treasuries
6.20 percent
Suppose value of the call feature is estimated to be 2.10 per $100 face value. That is, if it were
not callable the same bond would be expected to sell at a price of 92.00 + 2.10 = 94.10.
Option adjusted yield (at P = 94.10)
= 8.12 percent
Option-adjusted spread (OAS) = 8.12 - 6.20 = 1.92 percent
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Mortgages
Mortgages and Mortgage-Backed Securities
In the U.S., a mortgage loan is like a long term bond, except
• it is collateralized by the value of the house
• the principal is gradually repaid over the whole life of the loan, rather than in a single lump sum
payment at maturity.
There is an enormous total volume of mortgage debt outstanding: over $ 13.5 trillion by
2015.
Problems with mortgages
• Mortgage loans are illiquid (Loans are small, costly to service, and closely tied to the value of
•
the property and specific characteristics of the borrower.)
The homeowner always has the right to pay off the loan early. This makes mortgage loans
effectively callable at any time (which is a bother for the lender).
Mortgages are often pooled and pass-through securities, like GNMA's, and other mortgagebacked securities (MBS), are issued against the pool. These represent a different and
extremely important new class of derivative instrument. By the end of 2008, more than $7.5
trillion of the outstanding mortgage loans were held in pools underlying mortgage-backed
securities. Since the market meltdown in 2008, securitization of mortgages has gone way
down. By mid-2012 only $2.9 trillion was outstanding.
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Mortgages
Prepayment Risk
The future cash flows from a pool of mortgage loans depend on the prepayment
experience. Prepayment is hard to predict because it will depend on whether
interest rates go up or down in the future and also on "noneconomic" factors.
Noneconomic factors include
•
•
•
•
people move
borrowers default
transactions costs affect the refinancing decision
"nonrational" reasons, such as lack of information, may cause suboptimal
prepayment behavior
New types of derivatives were created to manage the impact of prepayment risk.
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Mortgages
Prepayment Risk and Valuation
Prepayments depend on the path taken by interest rates, so the value of the option
in a mortgage-backed security becomes path-dependent.
•
•
Prepayments for economic reasons (refinancing to get a lower interest rate)
increase when market interest rates fall below the rate on the mortgage
This effect is strongest the first time market rates fall to a new level; if they then
bounce up, the next time they fall to the same level there will be fewer prepayments
because the most interest-sensitive borrowers will have already prepaid. This is
called "burnout."
Therefore to price the mortgage or mortgage-backed security properly, you need
to know not just the current interest rate, but the entire past history of rates since
the security was issued.
Path dependence requires simulation methods (e.g., "Monte Carlo simulation")
to compute theoretical values for mortgage-backed securities.
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Mortgage-Backed Securities
Mortgage Loans
Banks make individual mortgage loans.
•
•
•
Expertise in evaluating property in the local market
Low default risk, because the house is collateral
High servicing requirements for the loans produce income for the lender (i.e.,
higher interest rate than on a bond with comparable risk)
Individual Mortgage Loans
Single
Loan
Single
Loan
Single
Loan
Single
Loan
Single
Loan
Single
Loan
But mortgage loans are illiquid and prepayment risk is hard to manage well.
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Mortgage-Backed Securities
Mortgage Pass-Throughs
Following the "Credit Crunch" of 1966, the Government National Mortgage Association
(GNMA, known as "Ginnie Mae") was created in 1970 to provide a new mechanism for
financing mortgage loans.
• The new financing idea is known as securitization.
• The new financial instrument was the mortgage pass-through security.
Mortgage lenders "originate" loans (they set them up). Once a number of similar mortgage
loans have been made, they are bundled together into a mortgage pool and GNMA passthrough securities are issued and sold in the securities market.
A mortgage pass-through is similar to a bond. Each month, as the homeowners make their
mortgage payments,
• the originating bank retains a fee for servicing the loan
• the rest of the funds are passed through to the holders of the pass-through securities
• to be eligible for inclusion in a GNMA pool, a mortgage loan must be insured by
the government (e.g., the Veteran's Administration). There is no risk to the lender
from a borrower default.
• prepayments of principal from loans paid off early by borrowers or from
government payoffs of defaulted loans are also passed through. This makes
monthly cash flow irregular and hard to predict.
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Mortgage-Backed Securities
Mortgage Pass-Throughs
Cash flows from the mortgage pool are allocated
proportionally to the pass-through securities
Mortgage Pool
pass-through security
Single
Loan
Single
Loan
Single
Loan
Single
Loan
pass-through security
Single
Loan
Single
Loan
pass-through security
pass-through security
The GNMA pass-through revolutionized the mortgage market. Funding of home loans no
longer depended on the ability of savings banks and S&Ls (savings and loan institutions) to
attract deposits: funds could be obtained as needed in the bond market.
But prepayments still create substantial risk for the investor.
• Cash flows are less predictable than with regular bonds
• Prepayments increase when interest rates are low, to the disadvantage of the lender
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Mortgage-Backed Securities
Collateralized Mortgage Obligations (CMOs)
Before long further innovations were introduced. More complex structures called CMOs
were introduced, with different "tranches" (French for "slice") of securities.
Each month when the homeowners make their mortgage payments
• holders of the "A" tranche get their promised payments first--these are extremely predictable
• then "B" tranche holders are paid from the remaining funds, and so on
• uncertainty in the total cash flows from the pool due to prepayment risk is concentrated in the
lower priority classes
Cash flows from mortgage pool are
allocated to different priority classes
(tranches) of mortgage-backed securities
Mortgage Pool
A Tranche
Single
Loan
Single
Loan
"waterfall"
Single
Loan
Single
Loan
Single
Loan
Single
Loan
B Tranche
C Tranche
Z Tranche
"toxic waste"
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Mortgage-Backed Securities
Pricing MBS using Monte Carlo Simulation
The payoffs on basic forwards, futures and European options depend on the price of the
underlying asset at the expiration of the contract. It doesn't matter how the price gets from
today's value to the final value. The payoff is independent of the path.
American options are different because you might want to exercise early, depending on
where the asset price is on one or more dates before expiration. The option is pathdependent, but only in a limited way. At each date you can decide whether to exercise or
not, but that decision will depend only on how the price might evolve from that date
forward. It doesn't matter how it got to its present level.
Mortgages and mortgage-backed securities are path-dependent in a more complicated way.
The timing and amounts of their payoffs are a function of the prepayment experience on the
underlying mortgage loans. It does matter what path interest rates have followed in getting
to today's level. That makes it impossible to price MBS with the standard backward
recursion technology we have considered so far.
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Mortgage-Backed Securities
Pricing MBS using Monte Carlo Simulation, p.2
Consider an MBS that is backed by a pool of mortgage loans with a 6% fixed interest rate.
We may know that the rate in the market is 6.5% today, but that's not enough to value the
MBS. Its future cash flows will be affected by how many of the underlying mortgages have
already prepaid, and that depends on how interest rates have moved before now.
If the rate has remained above 6% since the pool was formed, there may have been
comparatively few prepayments. But if rates have dropped to 4% and then gone back up to
today's 6.5%, many of the original borrowers will have prepaid their 6% loans and
refinanced at lower rates. The number of mortgages left in the pool will be much lower.
Path-dependent securities like MBS have to be priced approximately, by Monte Carlo
simulation.
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Mortgage-Backed Securities
Pricing MBS using Monte Carlo Simulation, p.3
Monte Carlo simulation starts with a model, like one of the interest models we have looked
at, for how the underlying risk factor(s) behave.
dr = κ ( µ - r ) dt + σ dz
Interest rate model:
rt +1 − rt =κ (µ − rt ) ∆t + σ z t ∆t
Discretized version:
A possible path for future interest rates {r1, r2, ..., rT} is simulated by drawing random
numbers from a standard normal distribution (or whatever distribution one thinks is most
appropriate) and plugging them into the equations for the stochastic {zt} terms. Once a full
path of rates from the present to maturity is generated, a prepayment model is used to
compute expected prepayments from the pool along that interest rate path.
Vi = V({r1, r2, ..., rT}i , prepayments)
The present value of the simulated cash flows along the ith simulated interest rate path gives
one observation, Vi , for the possible value of the MBS consistent with the interest rate
model and the assumed prepayment behavior.
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Mortgage-Backed Securities
Pricing MBS using Monte Carlo Simulation, p.4
This process is then repeated, maybe N = 100,000 times. In the end, one has 100,000
possible outcomes that are consistent with the interest rate process and the prepayment
model. These are used to compute the mean, standard deviation, and other necessary
statistics for the returns distribution, from which the fair value V and risk parameters are
obtained.
V = mean( {Vi, i = 1,...,N} )
Monte Carlo simulation is heavily used for solving models in:
• weather forecasting
• nuclear weapons design
• financial derivatives
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Credit Risk: Default as an Option and Credit Derivatives
Up to this point, our focus has been on hedging and managing exposure to the risk of adverse
fluctuations in the market values of our assets or liabilities. But for many financial institutions
like banks, the risk that worries them most is not that the present value of the repayments on a
loan will go down, but that a borrower will not repay the loan at all, i.e., credit risk.
An early insight from modern option valuation technology was that securities issued by a
corporation, like stocks and bonds, are actually derivatives whose values are derived from the
value of the underlying firm. Option pricing theory can help us understand the risk of
bankruptcy embedded in corporate securities.
Thinking of the stock as an option on the firm can help us value risky debt and also clarifies
how limited liability gives shareholders the incentive to increase risk exposure at the firm
level. The current "structural" and "reduced form" approaches to evaluating credit risk have
developed from this insight.
Finally we take a look at two important new types of derivative instruments that have been
developed specifically for managing credit risk: credit default swaps (CDS) and
collateralized debt obligations (CDOs).
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Credit Risk: Default as an Option and Credit Derivatives
Models of Default Risk
There are two basic derivatives-related approaches to analyzing and valuing risky debt (i.e.
debt with a risk of default): "Structural" models (based on ideas first published in a paper
by Merton in 1974), and "Reduced Form" models.
Structural Models
•
The true underlying asset is the whole firm, with current value Vt.
•
Vt follows a Black-Scholes type diffusion process
•
Bonds with face value F, maturing at date T, are paid off if VT > F
•
If VT < F, the firm defaults and the bondholders take over the firm (they get VT)
•
This makes equity a kind of call option on the underlying firm value V
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Credit Risk: Default as an Option and Credit Derivatives
The Structural Approach: Corporate Securities as Options on the Firm
The ability to default is an option
Consider a firm with a very basic capital structure. It has issued stock and a single zero
coupon bond.
Let
V = Value of the entire firm.
F = Face value of bonds to be paid off at date T.
V is assumed to follow the standard lognormal diffusion process
dV
V
= µ dt
+ σ dz
Consider the payoffs on the different securities on date T as a function of the firm value on
that date, VT.
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Credit Risk: Default as an Option and Credit Derivatives
Default in a Structural Model
Value of
Firm
V0
Face value
of debt F
Firm defaults here
time
T
Bond maturity date
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Credit Risk: Default as an Option and Credit Derivatives
Security Payoffs at Bond Maturity, Date T
VT < F
F < VT
1. STOCK
0
Firm is bankrupt
VT - F
Firm is solvent.
Bonds are paid in full.
2. BONDS
VT
Firm is bankrupt.
Assets distributed
to bondholders
F
Firm is solvent.
Bonds are paid
off at face value.
Note: The bond payoff is the same as (yet another example of put-call parity!)
Default-free bond
+
Writing a put option
on the firm value V
with strike price F
F
F
- (F - VT )
0
VT
F
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Credit Risk: Default as an Option and Credit Derivatives
Implications of Modeling Stock and Bonds as Contingent Claims
The stock is an option on the firm and option value is enhanced by higher volatility.
• The shareholders have an incentive to increase firm risk. (This shifts value from
the bondholders to the stockholders.)
• One way to increase risk is to distribute firm assets to the shareholders (paying
dividends); this is typically restricted by covenants in the bond indenture
agreement.
• but sometimes other opportunities arise to transfer firm value from the bondholders
to the stockholders, e.g., RJR Nabisco in the late 1980s (see the book and movie
"Barbarians at the Gates")
This approach is one of the major ways of addressing credit risk on bonds within the
standard contingent claims valuation paradigm.
• KMV (now Moody's KMV) is major firm currently doing credit analysis based
largely on the structural framework.
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Credit Risk: Default as an Option and Credit Derivatives
Problems with Structural Models of Default Risk
Firm value is hard to determine for real-world firms
Actual debt contracts are much more complicated than what Merton models
• "bonds" are coupon debt, with many small periodic payments before maturity, call
provisions and other special features
• firms may have issued multiple classes of debt with different maturities and other
terms (senior debt, junior debt, bank loans, lines of credit, commercial paper, etc.,
etc.)
• priority rules for who gets paid off first are often violated in real world bankruptcy
proceedings
• it is difficult to model the firm's optimal default strategy
The structural approach is little help in valuing many popular kinds of credit derivatives
whose payoffs are tied to a change in bond rating or yield spread, not to actual default
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Credit Risk: Default as an Option and Credit Derivatives
Reduced Form Models of Default Risk
The other major class of default models does not try to look carefully inside the corporation,
but just focuses on overall probabilities. These are known as "reduced form" models.
•
Default is a random event, like a lightning strike, that could happen to any firm at
any time.
•
Default is modeled as a Poisson process (a probability model for infrequently
occurring big events; almost the exact opposite of a diffusion process).
•
Default intensity (the probability of a default within a given span of time) can be
modeled as exogenous, or as a function of firm value and other variables.
•
Payoff on bonds if default occurs is not VT, but some exogenously specified
recovery rate (empirical recovery rates are quite variable, but average around 40 50%).
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Credit Risk: Default as an Option and Credit Derivatives
Moody's KMVHistorical Credit Ratings Transition Matrix
Source: Moodys KMV. "Default and Recovery Rates of Corporate Bond Issuers, 1920-2015."
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Credit Risk: Default as an Option and Credit Derivatives
Beginning in the mid-1990s, derivatives based on credit risk began to appear. The simplest
is the credit default swap (CDS), a derivative contract that functions very much like an
insurance policy against default by a bond issuer.
The collateralized debt obligation (CDO) uses securitization and tranching, as in
mortgage-backed securities, to reallocate the incidence of the default risk in a portfolio of
risky bonds. An MBS concentrates prepayment risk into a small fraction of the new
securities, leaving the rest with virtually none. In the same way, a CDO concentrates default
risk into a small fraction of the CDO tranche securities, leaving most of the tranches at AAA
quality or above, even when the underlying bonds in the pool are much riskier.
Unfortunately, valuation models for credit derivatives are complicated and hard to test
empirically because defaults are such rare events (luckily!). Pricing in the real world does
not seem to be entirely consistent with the theoretical models.
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Credit Risk: Default as an Option and Credit Derivatives
Credit Default Swaps
A CDS is a derivative contract based on default risk. Like a futures contract, a CDS
transfers risk from the owner of the risky security to the derivatives counterparty.
How it works (originally):
Counterparty A (the protection buyer) commits to make regular premium payments to
Counterparty B. The rate is S basis points per year, applied to a notional face value F.
Counterparty B (the protection seller) commits to make the following payments to A:
• If there is no default by the "reference entity" (the bond issuer) before the CDS
expires, B pays nothing.
• If the reference entity defaults, B must compensate A's loss. Either
° A delivers bonds issued by the defaulting entity, and B pays A their face value F, or
° B pays a cash amount to A, equal to the difference between the post-default price of the
bonds and face value F. The price is determined in a special auction 1 month after the
credit event.
Market conventions changed in 2009: The premium payments are now standardized, so most CDS
pay 100 basis points annually (high risk CDS pay 500 b.p.), and an upfront payment adjusts for the
precise level of default risk. If equilibrium spread S for a particular issuer would be below 100 b.p., the
protection seller will be receiving too much premium, so the seller pays the buyer the fair value of the
difference up front. If equilibrium S would be above 100 b.p., the buyer pays the seller up front.
Maturity dates are also standardized. Both of these changes are big improvements to liquidity in the
secondary market for CDS.
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Credit Risk: Default as an Option and Credit Derivatives
An Extremely Important Innovation: The Credit Default Swap
For many institutional investors, market price risk is much less important than risk of
default. The CDS is a way to buy (or sell) insurance against default.
The protection buyer pays a regular quarterly premium to the protection seller. If there is a
default, the protection seller must pay the protection buyer. The amount of compensation
that is determined at a special auction of the defaulted bonds about a month later.
Payment of interest and principal +
loss of principal in case of default
Total Alpha Fund
buys protection CDS premiums;
on bonds
plus defaulted
issued by
bonds, if any
XYZ Corp.
Reimbursement
of loss (if any) on
defaulted bonds
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Financial
Markets
Credit default
swap
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Credit Risk: Default as an Option and Credit Derivatives
An Extremely Important Innovation: The Credit Default Swap
In the previous example, the Total Alpha Fund bought protection to eliminate the default risk
on XYZ Corp. bonds, to produce a portfolio with credit quality better than AAA.
Alternatively, Total Alpha could hold a portfolio of Treasury bonds and sell protection on
XYZ. It would then receive regular premium payments from its counterparty, but would bear
the risk of loss if a default occurs. This way Total Alpha earns the return, including risk
premium, on risky XYZ bonds (and bears the default risk) without actually owning them.
Return on Treasury bonds
Total Alpha Fund
Financial
Markets
CDS premiums
Reimbursement
of losses if
XYZ defaults
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Credit Risk: Default as an Option and Credit Derivatives
CDS Pricing
The equilibrium CDS spread is the value of the spread S that sets the (risk neutral) expected
values of the "premium leg" and the "protection leg" equal. For a T period CDS,
Let
pi = probability of default in period i, for a bond that starts at time 0
Di = the discount factor for premium payment i
θi = the length of the payment period
R = fraction of face value recovered in case of default
V0 = upfront payment by protection buyer (negative if seller pays upfront)
 i −1 
D i θi 1 − ∑ p j 
∑
i =1
j=1


Premium Leg
E[PV(future stream of premiums)] = V0 + S
Protection Leg
E[PV(payoff in case of default)] =
T
probability of
no default prior
to period i
T
(1 − R) ∑ Di pi
i =1
Note: Under the new procedures, S is always set at 100 (or 500). The two legs will not have
the same value. The difference between them is the upfront payment that will be needed.
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Credit Risk: Default as an Option and Credit Derivatives
The Equilibrium CDS Spread S versus the Credit Spread in Bonds
The equilibrium CDS spread should be close to the credit spread on the firm's bonds in the
bond market. But they are not equal in practice, and there are some practical reasons why
they should not be equal.
•
There is an option of which of several bonds to deliver against the CDS
•
At the time of default, accrued premium since the previous payment date must be
paid by the protection buyer, but the bondholder doesn't get any accrued interest
•
Both corporate bonds and CDS are subject to market noise that reduces measured
correlation between them (corporate bonds are pretty illiquid)
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Credit Risk: Default as an Option and Credit Derivatives
The Recovery Rate Assumption
•
The recovery rate assumption has a very significant impact on CDS pricing.
•
The recovery rate is hard to predict accurately. It is commonly simply set to the
long run average recovery fraction, approximately 40%.
•
Average recovery rates have been found to vary over a wide range across different
defaults and over time
•
recoveries are negatively correlated with (physical) default probabilities (the more
likely default is, the less will probably be recovered if default happens)
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Credit Risk: Default as an Option and Credit Derivatives
Collateralized Debt Obligation (CDO)
CDS (Credit Default Swaps) are like a combination of futures and insurance. The primary
purpose is to transfer exposure to risk
CDOs (Collateralized Debt Obligations) are a different class of derivatives, like mortgagebacked securities. The primary purpose is to repackage risk exposure
A CDO is a securitization of debt securities
•
•
•
•
risky bonds, or loans, are pooled and new securities similar to CMOs are issued,
with different priorities over the cash flows
CDOs can concentrate the default risk of the underlying bonds into a few high-risk
securities, leaving the others essentially risk free
"synthetic CDOs" are pools not of risky bonds, but of CDS
one of the most important properties of the pool is the correlation in default risk
across issuers (which determines the risk that a lot of bonds will go bad together)
Originally, most mortgages in the pools supporting mortgage pass-throughs and CMOs were
guaranteed by the government, so there was no credit risk. In the years following 2000, a
large volume of CDOs based on pools of "subprime" and "Alt-A" mortgage loans were
created. These are not guaranteed, so the CDOs were exposed to both prepayment risk and
default risk. In a number of cases the underlying mortgages were so bad that the tranching
failed to protect the senior tranches.
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Credit Risk: Default as an Option and Credit Derivatives
Collateralized Debt Obligations
Securitization and tranching used in creating mortgage-backed securities can be applied to
other kinds of loans (auto loans, credit card receivables, etc., etc.).
Some of the most exciting new securitized instruments are Collateralized Debt Obligations
and Collateralized Loan Obligations (CLOs)
• pools of credit-risky bonds or loans are bundled together ("cash flow CDO")
• a common alternative structure is the "synthetic CDO," in which the pool of securities exposed
to credit risk consists of Credit Default Swaps, rather than actual bonds
Pool of Bonds or Loans
Single
Loan
Single
Loan
Supersenior
Single
Loan
Single
Loan
Single
Loan
Single
Loan
Senior
Tranche
Mezzanine
Tranche
Equity
Tranche
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Credit Risk: Default as an Option and Credit Derivatives
Allocation of Default Losses in CDOs
In a CDO, default losses are allocated first to the lowest tranche
• the "senior" and "super-senior" tranches are typically more than 80% of the total principal value
• these can be rated AAA, even though the underlying bonds or loans are rated much lower.
Pool of Bonds or Loans
Single
Loan
Single
Loan
Supersenior
Single
Loan
Single
Loan
Single
Loan
Single
Loan
Senior
Tranche
Mezzanine
Tranche
Equity
Tranche
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Credit Risk: Default as an Option and Credit Derivatives
Allocation of Default Losses in CDOs
The price for the equity tranche is quoted differently from the other tranches in the market
because it has a relatively high probability of being completely wiped out before the pool
matures
Pool of Bonds or Loans
Single
Loan
Single
Loan
Supersenior
Single
Loan
Single
Loan
Single
Loan
Single
Loan
Senior
Tranche
Mezzanine
Tranche
Equity
Tranche
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Credit Risk: Default as an Option and Credit Derivatives
CDOs
The essential features of a CDO are securitization and tranching.
Typical structure
Tranche
"super senior"
"senior"
"mezzanine"
"equity" or "first loss"
Range of Losses Covered
above 12%
7 - 12%
3 - 7%
0 - 3%
Just like with a CMO, the cash flows into the portfolio, either from bond coupon and principal
payments (or from CDS premium received in a synthetic structure) are passed through and
allocated to the tranches, in a "waterfall" pattern.
If there is a default, the principal backing the equity tranche is reduced by the amount of the
loss. All default losses are allocated to the equity tranche until the total loss is greater than 3%
of the initial principal and the tranche is totally wiped out. Any defaults after that will be borne
by the mezzanine tranche, until 7% of the face value has been wiped out. (Note that because
there are recoveries, this would mean that a lot more than 7% of the bonds have defaulted.)
The super senior tranche is insulated against all defaults on the portfolio that do not total more
than 12% of initial principal. This is extremely unlikely unless defaults are highly correlated.
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Credit Risk: Default as an Option and Credit Derivatives
Correlation and CDO Valuation
One of the major determinants of default risk exposure in a CDO is correlation in defaults.
A simple example:
• The "portfolio" = 50% in bonds from issuer A and 50% in bonds from issuer B.
• Both A and B have 10% probability of defaulting within the next year.
• There are two CDO tranches issued, each covering 50% of face value.
The equity tranche (0 - 50%) takes the first loss. It has a total loss if either A or B (or both)
defaults. The senior tranche only takes the second loss, so there is no loss to the senior
tranche unless both A and B default.
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Credit Risk: Default as an Option and Credit Derivatives
Correlation and CDO Valuation, p.2
Impact of correlation
If ρ = 0, defaults are independent
• prob(loss to equity tranche) =
A&B bonds
10% default with
zero correlation
A: bonds
10% default
B bonds
10% default
p(A) + p(B) - p(A and B) = 19%.
•
prob(loss to senior tranche) = p(A) × p(B) = 1%.
9% prob A bonds
default and B bonds
don't
1% chance both
A and B bonds
default
9% prob B bonds
default and A bonds
don't
If ρ = 1.0, defaults for both A and B occur together
• prob(loss to equity tranche) = p(A) = p(B) = 10%.
• prob(loss to senior tranche) = p(A) = p(B) = 10%.
A&B bonds
10% default with
perfect correlation
The basic result: Higher default correlation increases
the value of the equity tranche and reduces the value
of the senior tranche.
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10% chance A and
B bonds all default
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Credit Risk: Default as an Option and Credit Derivatives
Other Credit Derivatives
First-to-default, nth-to-default basket credit default swap:
•
The option payoff is based on default experience of a portfolio of reference entities
(risky issuers). Like a single-name CDS, it pays off when there is a default, but
only after there have been n-1 previous defaults in the group (n may be 1, making
the first default trigger the payoff).
Total return swap:
•
•
Counterparty A commits to pay Counterparty B the total return on some asset (in
this case, a risky bond, or portfolio of risky bonds), including the loss of principal
in case of a default
B commits to pay A the total return on some default-free security (e.g., 10-year
Treasuries)
CDO-squared, CDO-cubed (nearly extinct):
•
A Collateralized Debt Obligation for which the underlying pool that is being
tranched is made up of CDO tranches. For example, a CDO-squared could be
based on a pool of the 3-7 mezzanine tranches of CDOs. The use of the term has
evolved over time, so that "CDO" is often used for the CDO-squared structure,
while a CDO of whole loans, not tranches, is called an ABS (asset backed security).
The experience of 2008 showed that CDO-squared and cubed structures don't work.
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Credit Risk: Default as an Option and Credit Derivatives
Conclusions on Credit Risk and Credit Derivatives
Default risk is important. And interesting.
Credit derivatives are a major innovation for our financial system.
Implied default probabilities and correlations measure risk neutral values. We are a long
way from fully understanding them, or from extracting dependable information about true
probabilities from them.
It is very hard to test our models of credit risk, because defaults are very rare events. The
world provides us with data on them very slowly. Fortunately!
We have had relatively little experience with how these instruments actually work when
there are substantial numbers of defaults. Early credit events with CDS led to significant
revision of the contract terms. The sub-prime mortgage crisis has led to changes in CDO
structures.
One major effect of the 2008 crisis has been that the securitization industry remains
essentially dead, except for securities issued by the quasi-government agencies Fannie Mae
and Freddie Mac.)
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Exotic Options, Real Options, and Structured Products
In this final session, we look at options with more complex payoffs than the "plain vanilla"
calls and puts we have considered so far. Some of these are nothing more than packages of
standard options, while others are quite different, presenting both new payoff patterns and also
new valuation problems, some of which become quite intractable.
We then consider two of the important "frontier" areas in applying derivatives concepts in the
real world: Real Options and Structured Products.
One direction in which option theory has been extended is to aid in real investment decisions.
Given enough assumptions, contingent claims theory can place a specific dollar value on the
element of choice in an investment project, such as the ability to shut it down prematurely if it
is unsuccessful, or to expand its scale if it does better than anticipated.
Another direction is toward the use of "financial engineering" to create tailor-made financial
instruments designed for a specific purpose. For more complicated structures, this may
involve creating a "Special Purpose Vehicle," which is a separate financial entity set up to hold
some kind of existing security and to issue various types of derivatives against it.
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Exotic Options, Real Options, and Structured Products
Exotic Options
There is an infinite range of possibilities in designing option payoffs. This has led to a
remarkable proliferation of "exotic" options. Most serve a real need, despite appearing very
strange to an outside observer, in some cases.
We may distinguish four families of exotic options, based on the techniques needed to value
them:
Path-independent exotics: These can be valued easily using the standard methodology,
such as the Binomial model. Some have closed form valuation equations.
Path-dependent exotics with path-independent valuation: Like American options,
these securities have payoffs that depend on the path followed by the underlying asset, but
they can still be priced with standard methods.
Path-dependent instruments without path-independent valuation: For these, the path
followed by the underlying determines the payoff in a way that requires different valuation
techniques, such as Monte Carlo simulation.
Multivariate options: The payoff is a function of more than one random variable.
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Exotic Options, Real Options, and Structured Products
Path-Independent Exotic Options
The key characteristic is that they can be valued knowing only the current price of the
underlying, not the price path up to this point in time.
Examples
Package: Complex payoff structures created by packages of simple options include
spreads, straddles, collars and range forward contracts. Valuation is easy: just sum up the
values of the component options.
Binary or Digital Option: Pays off a fixed amount if the underlying price at expiration is
above (call) or below (put) the option strike, no matter how far the option is in-the-money.
Valuation is easy with standard methods, but hedging is hard when the underlying is close to
the option's strike price near expiration.
Compound Option: An "option on an option," is the right to buy (call) or sell (put) an
underlying option. This is a useful conceptual device for modeling certain kinds of
compound contingencies, such as the value of an option on a stock in a firm subject to
bankruptcy risk. Valuation is not hard with standard methods.
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Exotic Options, Real Options, and Structured Products
Path-Dependent Exotic Options with Path-Independent Valuation
Some path-dependent contracts can still be valued with standard methods, if the value only
depends on the paths the underlying asset price might follow in the future. In other cases,
like barrier options and lookbacks, a valuation equation exists because under risk-neutral
valuation there is a closed-form expression for the expected value of the path statistic that
determines the payoff (for example, there may be a formula for the expected value of the
maximum price for the underlying over the option's lifetime).
American Options: An American option is exercised early the first time the price of the
underlying asset reaches the early exercise boundary. This makes the option pathdependent, but it can be valued using standard tools like the Binomial model.
Barrier ("Knock-in" and "Knock-out") Options: The payoff depends on whether the
price of the underlying hits a specified barrier level at some point during its lifetime. "In"
options, such as a "down and in call," must reach the barrier to become activated. If the
asset price does hit the barrier (called the "in strike"), the payoff at maturity will be the same
as a European option; otherwise the option expires worthless regardless of the asset price at
expiration. "Out" options must not hit the barrier ("out strike"); if the barrier is breached, the
option is knocked out and becomes worthless.
Lookback Options: A lookback pays off at maturity based on the highest or lowest price
the underlying asset reached during the option's life. For example, a lookback call pays the
difference between the final asset price and the lowest price observed over the option's
whole lifetime.
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Exotic Options, Real Options, and Structured Products
Path-Dependent Exotic Options without Path-Independent Valuation
Asian Options: Payoff is based on the average price of the underlying over its lifetime (or
over some specified portion of its lifetime). This payoff pattern is useful to manage risk
exposures that are themselves averages, such as the average monthly cost of natural gas
during the wintertime. It also reduces the incentive to try to manipulate the market price of
the underlying to affect the payoff at option expiration.
Valuation is not possible with standard methodology because the average price of the
underlying as of option expiration day is a function of the whole path followed by the asset
price up to that point, and not just the final price (and the arithmetic average of lognormal
variables is not lognormal). A variety of approximation formulas exist, or valuation may
also be done using Monte Carlo simulation of paths.
Mortgage-backed Securities: Prepayments on the underlying mortgages are pathdependent, so a mortgage-backed security cannot be valued without taking into account the
entire previous history of interest rates. Valuation is done using Monte Carlo simulation of
interest rate paths and prepayments.
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Exotic Options, Real Options, and Structured Products
Multivariate Options
Some exotic contracts depend on the price of more than one asset. In some cases, the
problem can be redefined to involve just one state variable, but usually not.
Exchange Options: The option to exchange asset X for asset Y clearly depends on the
values of both X and Y, but only on their difference (payoff = Max(Y- X, 0)). For this case,
a simple Black-Scholes-type formula exists (Margrabe, 1978).
Rainbow Options: These are often called an "option on the max" (or on the "min"). A
call on the max gives the holder the right to pay the strike price and acquire the best
performing among a set of underlyings. For example a call on the max might pay the
realized return on the S&P 500 stock index or on a 30 year U.S. Treasury bond, whichever
was greater. Closed-form valuation equations may exist, but are too complicated to be
usable except for 2-color, or at most 3-color, rainbow options.
Quantos: The underlying is denominated in one currency, but the payoff is in a different
currency. These are surprisingly common. An example is a call option based on the change
in the level (in Yen) of the Japanese Nikkei stock index, with the payoff being made in U.S.
dollars.
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Exotic Options, Real Options, and Structured Products
Real Options
An important extension of option pricing theory is toward "real" options. Real options
represent options--choices--that may be available in the future, for example, in a real
investment project. Optionality can contribute significant value to a project. Option theory
provides a framework for analyzing those choices and placing an economic value on them in
an investment decision.
Examples of real options include:
•
•
•
•
•
the option to initiate a new project or to abandon an existing one (e.g., the option of
when and how extensively to begin drilling in a new oil field; the option to shut
down an investment project early if it turns out to be unprofitable)
flexibility to change the scale of a project (e.g., the value of a factory design that
easily allows production to be expanded or reduced)
timing options (e.g., the option to speed up or slow down production; the option to
suspend production temporarily)
flexibility in product or input mix (e.g., the ability to switch a power plant from
coal to natural gas)
etc.
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Exotic Options, Real Options, and Structured Products
Real Options
The concept of real options helps frame consideration of future choice possibilities in an
investment project, simply by recognizing some of the portfolio dominance properties that
must hold. For example,
•
•
•
•
as a kind of option, the ability to make a choice in the future has positive value
the greater the volatility (uncertainty), the more the real option is worth
the longer the period before the choice has to be made, the more the option is worth
etc.
Placing a specific dollar figure on a real option requires assumptions (whose validity may be
questionable, such as that the "underlying" follows a lognormal diffusion) and also a pricing
model.
One big issue is that the "underlying" (e.g., expanded production capacity) is seldom an
investible instrument, so there is no arbitrage between the underlying and the option.
Pricing real options must be by equilibrium principles, rather than arbitrage, which requires
putting a "price of risk" on every stochastic factor in the problem.
Real option theory is still very much under development. But just recognizing that the
ability to make a choice in the future has tangible economic value which should be weighed
in any investment decision, and a conceptual framework to estimate that value, are already
major innovations in practical financial decision making.
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Exotic Options, Real Options, and Structured Products
Structured Products
Structured products are tailor-made financial instruments, often containing a variety of
derivative features, and designed for a specific purpose.
Structured products are created for a number of reasons, including:
• to manage the incidence of undesirable types of risk or other characteristics of the
underlying asset (e.g., prepayments, credit risk, ...)
• to create liquidity for an illiquid class of assets
• to secure more favorable tax or regulatory treatment
• to avoid unfavorable accounting treatment
Often, a new financial firm (a "special purpose vehicle" or "SPV") will be set up to hold the
underlying collateral and to issue the derivative securities.
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Exotic Options, Real Options, and Structured Products
A Sophisticated Structured Deal: Morgan Stanley's "PLUS I" Program
The Problem: Banamex, a major Mexican bank was holding a large amount of an inflationlinked bond issued by the Mexican government, known as "Ajustabonos." There were a
number of interlinked difficulties with these:
•
•
•
•
They were illiquid. Inflation had gone down, so there was little demand for them.
Prices had fallen, but they were being carried on the bank's books at face value.
Selling them would require recognizing a large loss.
They were denominated in pesos, so a US or other non-Mexican investor would
have a sizable exchange rate risk in owning them.
Mexican government bonds denominated in pesos had very little default risk, but
dollar-denominated government debt had a low bond rating.
Banamex wanted to "sell" these bonds to use the funds tied up in them for more productive
purposes, but
• There was no demand for them from Mexican investors,
• US investors wouldn't buy them unless they were highly rated and paid off in
dollars, and
• Actually selling them would entail booking an unacceptable loss on the bank's
accounting statements.
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Exotic Options, Real Options, and Structured Products
Morgan Stanley's "PLUS I" Program
The Solution: Securitization.
Use the Ajustabonos as the collateral to support the issuance of new bonds. A large tranche
of the new bonds would be designed to have the features that US investors needed. And just
like the individual mortgage loans in a mortgage pool that support a set of collateralized
mortgage obligations, the Ajustabonos would provide the cash flow for the coupon interest
and principal payment of the new bonds.
How to accomplish all of this?
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Exotic Options, Real Options, and Structured Products
Morgan Stanley's "PLUS I" Program
1. To do the securitization, it is common to set up a new financial firm--a "Special Purpose
Vehicle" (SPV), wholly owned by the firm doing the deal. In this case, it was a Bermudabased corporation. Legal aspects of doing this were amusing but not very important.
2. Transfer the Ajustabonos to the SPV. The SPV then issued two classes of new bonds.
80% of the new bonds were denominated in US dollars and had higher priority than the
remaining 20% (i.e., they got paid first).
3. The lower priority bonds were retained by Banamex. These served as a cushion to
improve the credit quality of the senior bonds. If the peso/dollar exchange rate changed
adversely, the impact on the dollar-denominated bonds would be offset by reduction in the
payments to the bonds held by Banamex. The PLUS I bonds were therefore rated AA- by
Standard and Poor's, a high investment grade rating.
4. The bonds held by Banamex were counted as paid-in capital invested in the SPV. Since
the SPV was just a wholly-owned subsidiary of Banamex, transferring the Ajustabonos to it
did not count as a sale for accounting purposes, so no loss was recorded on Banamex's
books.
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Exotic Options, Real Options, and Structured Products
Morgan Stanley's "PLUS I" Program
Structured products are created for a number of reasons. This deal illustrates all of
them:
•
"to reduce the impact of undesirable types of risk or other characteristics of the
underlying asset (e.g., prepayments, credit risk, ...)"
In this case, it was exchange rate risk that was the problem.
•
"to create liquidity for an illiquid class of assets"
The illiquid Ajustabonos were turned into highly marketable PLUS I bonds.
•
"to secure more favorable tax or regulatory treatment"
Here the special treatment needed was an investment grade bond rating.
•
"to avoid unfavorable accounting treatment"
Banamex was able to bring in cash without actually selling the bonds and
booking an accounting loss.
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Exotic Options, Real Options, and Structured Products
Morgan Stanley's "PLUS I" Program
Partnoy is highly scornful of this deal. Let us think about the various parties involved and
ask whether they were hurt by it, helped by it, or not affected one way or the other.
How about:
• Morgan Stanley
• Banamex
• Bermuda school girls
• US buyers of PLUS I notes, like the Wisconsin pension fund
• The Mexican government, issuer of the Ajustabonos
• Mexican borrowers who are now able to get loans from Banamex
• other Mexican holders of Ajustabonos
Bottom line: It seems as if almost everyone benefited from this deal and no one was hurt.
The derivatives concept and some creative financial engineering produced a new structure
in the financial market that made it possible for capital to flow more freely and more
efficiently.
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Concluding Thoughts
A Few General Principles of Options
Buying options reduces overall risk exposure; writing options increases it
You pay for what you get.
•
•
If one position is better than another under one scenario, it will be worse under other
scenarios, and the market considers the tradeoff to be fair.
Writing out of the money options naked is extremely dangerous, even though they
nearly always end up out of the money. Writing out of the money options and delta
hedging them is also risky, but not quite as bad.
End users should use options to achieve desirable payoff patterns, not to exploit
"mispricing."
•
•
Options allow payoffs to be precisely tailored to match the investor's preferences
and market view.
Trading costs and risk are too large for a non-professional to profit from mispricing
relative to a theoretical model.
Many option strategies "work" for a particular investor because they alter a
position's tax or accounting treatment, or its liquidity.
FINC-UB.0043 Futures and Options Spring 2018
Part II: Option Pricing and Hedging; Advanced Topics
©2018 Figlewski
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Concluding Thoughts on the Derivatives Course
A Few General Principles to Carry Forward
Good luck on the final exam, and especially
BEST WISHES FOR THE SUMMER
AND FOR YOUR FUTURE CAREERS!!
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